1 Introduction
1.1 Discrete Chebyshev system
Let
$C[a,b]$
be the space of continuous real-valued functions on a finite interval
$[a,b]$
equipped with the norm
$ \|f\|_{\infty }=\max _{t\in [a,b]}|f(t)|. $
Given linearly independent functions
$\{\varphi _k(t)\}_{k=1}^n\subset C[a,b]$
, the set of polynomials is defined by
$$\begin{align*}L_n=\Big\{p=\sum_{k=1}^na_k\varphi_k\colon a_k\in\mathbb{R}\Big\}. \end{align*}$$
The best approximation of a continuous function f by
$L_n$
is given by
$$ \begin{align*} E(f,L_n)_{\infty}=\min_{p\in L_n}\|f-p\|_{\infty}=\|f-p^{*}\|_{\infty} \end{align*} $$
and
$p^{*}\in L_n$
is the best approximant.
The Chebyshev systems play the key role in a study of the best approximation of continuous functions by the n-dimensional subsets. Recall that a set of continuous functions
$\{\varphi _k\}_{k=1}^n$
is called a T-system (or a Chebyshev system) on
$[a,b]$
if any nontrivial polynomial with respect to this system has at most
$n-1$
distinct zeros on
$[a,b]$
. This definition can be equivalently written as follows (cf. [Reference Karlin and Studden21, Ch. 1, § 4]): The set
$\{\varphi _k\}_{k=1}^n$
is a T-system if and only if, for any sequence
$a\le t_1<\dots <t_n\le b$
, the determinants
have the same sign.
Let us recall two crucial facts on the best approximations in
$C[a,b]$
.
Haar’s theorem [Reference Laurent24, Th. 3.4.6]
The best approximant
$p^{*}\in L_n$
of a function
$f\in C[a,b]$
is unique if and only if
$\{\varphi _k\}_{k=1}^{n}\subset C[a,b]$
is a Chebyshev system.
Chebyshev’s theorem [Reference Laurent24, Th. 3.4.7]
Let
$\{\varphi _k\}_{k=1}^{n}\subset C[a,b]$
be the Chebyshev system. A polynomial
$p^{*}\in L_n$
is the best approximant of a function
$f\in C[a,b]$
if and only if there exists an ordered set of
$n+1$
points
$a\le t_1<\dots <t_{n+1}\le b$
such that
$$ \begin{align} \begin{aligned} \text{(i)}&\quad |p^{*}(t_i)-f(t_i)|=\|p^{*}-f\|_{\infty},\quad i=1,\dots,n+1,\qquad \\ \text{(ii)}&\quad p^{*}(t_i)-f(t_i)=-(p^{*}(t_{i+1})-f(t_{i+1})),\quad i=1,\dots,n. \end{aligned} \end{align} $$
Recall that an ordered set of
$\{t_i\}$
satisfying properties (1.2) is called the Chebyshev alternance set of length
$n+1$
.
We are interested in analogues of Haar’s and Chebyshev’s theorems for the best uniform approximation of discrete functions defined on ordered sets of points, rather than on an interval. Note that some basic results can be partially derived from well-known results on the best uniform approximation of continuous functions on an arbitrary compact set (see, e.g., [Reference Dunham11], [Reference Dzyadyk and Shevchuk12, Ch. 1, §§ 2, 5], [Reference Gantmacher and Krein15, Ch. II], [Reference Laurent24, Ch. 3]).
For
$m,n\in \mathbb {Z}$
,
$m\le n$
, we define
$[m,n]_{{\scriptscriptstyle \mathbb {Z}}}=\{m,m+1,\dots ,n\}$
. For
$q\in \mathbb {N}$
, letFootnote
1
$C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}=\{f\colon [0,q]_{{\scriptscriptstyle \mathbb {Z}}}\to \mathbb {R}\}$
be the linear space (of the dimension
$q+1$
) of real-valued discrete functions on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
equipped with the norm
$ \|f\|_{\infty }=\max _{\nu \in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}}|f(\nu )|. $
The point
$\nu $
is called a zero of the discrete function f ifFootnote
2
either
$$\begin{align*}f(\nu)=0\ \ \text{for}\ \ \nu \in [0,q]_{{\scriptscriptstyle\mathbb{Z}}} \quad \text{or}\quad f(\nu-1)f(\nu)<0\ \ \text{for}\ \ \nu \in [1,q]_{{\scriptscriptstyle\mathbb{Z}}}. \end{align*}$$
It will be important to distinguish these zeros and we will call them the zeros of the first or second type, respectively. Denote by
$N(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})$
the number of zeros (of both types) of f on
$[m,n]_{{\scriptscriptstyle \mathbb {Z}}}$
. Set also
$N(f)=N(f,[0,q]_{{\scriptscriptstyle \mathbb {Z}}})$
. Moreover, let
$N_0(f)$
be the number of zeros of the first type of f on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. Clearly,
$N_{0}(f)\le N(f)$
.
F. Gantmacher and M. Krein [Reference Gantmacher and Krein15, Ch. II] suggested the following characteristics of oscillatory properties of discrete functions. Let
$S^{-}(f)$
(respectively,
$S^{+}(f)$
) be the least (the largest) number of sign changesFootnote
3
of f on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
after replacing all zero values of f by arbitrary nonzero values. We claim that
$$\begin{align*}S^{-}(f)\le N(f)\le S^{+}(f), \end{align*}$$
with the discussion and proof given in Section 3 (see Lemma 3.1).
We study the problem of the characterization and uniqueness of the best uniform approximants of discrete functions. Let
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
be linearly independent discrete functions and
$L_n$
be its linear span over
$\mathbb {R}$
. As in the continuous case,
$$\begin{align*}E(f,L_n)_{\infty}=\min_{p\in L_n}\|f-p\|_{\infty}=\|f-p^{*}\|_{\infty} \end{align*}$$
is the best uniform approximation of f by
$L_n$
and
$p^{*}\in L_n$
is the best approximant.
Since, for discrete functions one can distinguish between two types of zeros, we define two corresponding discrete Chebyshev systems. Let
$n\in [1,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. The set of discrete functions
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is called the
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system (respectively,
$T_{0}$
-system) if for any nontrivial polynomial
$$ \begin{align} p(\nu)=\sum_{k=1}^{n}a_k\varphi_k(\nu) \end{align} $$
of degree at most n with real coefficients we have
$N(p)\le n-1$
(
$N_0(p)\le n-1$
). It is clear that any
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system is also
$T_0$
-system but the inverse statement is not valid, see the example in Section 2.
Let
$p^{*}\in L_n$
be the best uniform approximant of the function
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
and set
$$\begin{align*}e(\nu)=p^{*}(\nu)-f(\nu), \quad \nu \in [0,q]_{{\scriptscriptstyle\mathbb{Z}}}. \end{align*}$$
Due to compactness of
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}\subset \mathbb {R}$
, we observe that an analogue of Haar’s theorem is valid only for
$T_0$
-systems.
Theorem 1.1 [Reference Laurent24, Th. 3.4.6]
The best uniform approximant
$p^*\in L_n$
of the function
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is unique if and only if the set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_0$
-system.
We note that a (nontrivial) polynomial (1.3) vanishes at n integer points
$0\le \nu _1<\dots <\nu _n\le q$
if and only if the determinant
$$ \begin{align} \Delta\left( \begin{matrix} \varphi_1, & \varphi_2, & \dots, & \varphi_n\\ \nu_1, & \nu_2, & \dots, & \nu_n \end{matrix}\right) \end{align} $$
defined by (1.1) is zero. Therefore, we arrive at the following obvious statement, which provides a criterion for
$T_0$
-systems (see, e.g., [Reference Karlin and Studden21, Ch. VII, § 1]).
Proposition 1.2. The set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_0$
-system if and only if the determinants (1.4) are nonzero for any integers
$0\le \nu _1<\dots <\nu _n\le q$
.
A straightforward analogue of Chebyshev’s theorem for
$T_0$
-systems reads as follows.
Theorem 1.3 [Reference Laurent24, Th. 3.4.7]
Let the set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
be a
$T_0$
-system. The polynomial
$p^{*}\in L_n$
is the best uniform approximant of a function
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
if and only if there exist distinct points
$\nu _1, \dots ,\nu _{n+1}\in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\varepsilon _1,\ldots ,\varepsilon _{n+1}\in \{-1,1\}$
, and positive numbers
$\rho _1,\ldots ,\rho _{n+1}$
, satisfying the conditions
$$ \begin{align*} \begin{aligned} \mathrm{(i)}&\quad \varepsilon_i e(\nu_i)=\|e\|_{\infty},\quad i=1,\dots,n+1,\quad e=p^{*}-f,\\ \mathrm{(ii)}&\quad \sum_{i=1}^{n+1}\varepsilon_i\rho_i p(\nu_i)=0\quad \text{for all}\quad p\in L_n. \end{aligned} \end{align*} $$
We note that the set of points
$\nu _1,\dots ,\nu _{n+1}$
in Theorem 1.3 may not form the Chebyshev alternance set as we will see in Example 2.7.
Our first main result provides a characterization of the discrete set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
such that for any
$f \in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
the best uniform approximant
$p^{*}\in L_n$
has an alternance set of length
$n+1$
, that is, we obtain a complete analogue of Chebyshev’s alternance theorem.
We say that the best uniform approximant
$p^{*}\in L_n$
of f admits an alternance set
$0\le \nu _1<\dots <\nu _{n+1}\le q$
of length
$n+1$
if the following condition holds (cf. (1.2)):
$$\begin{align*}\varepsilon(-1)^{i}e(\nu_i)=\|e\|_{\infty},\quad i=1,\dots,n+1,\quad \varepsilon=\pm 1. \end{align*}$$
Theorem 1.4. Let
$n\in [1,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. The following are equivalent
$:$
-
(a) The set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system. -
(b) For any function
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, the best uniform approximant
$p^{*}\in L_n$
of f admits an alternance set of length
$n+1$
. -
(c) Determinants (1.4) constructed over all sets
$0\le \nu _1<\dots <\nu _n\le q$
are nonzero and have the same sign.
Thus, an analogue of Chebyshev’s theorem holds only for
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-systems.
Using Theorem 1.4, one can easily construct different examples of
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-systems via T-systems on an interval.
Corollary 1.5. If
$\{\varphi _i(t)\}_{i=1}^n\subset C[0,q]$
is a T-system on
$[0,q]$
, then
$\{\varphi _i(\nu )\}_{i=1}^n\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system.
Another important property of
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-systems is given below.
Corollary 1.6. If
$\{\varphi _k\}_{k=1}^n\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system, then for any nontrivial polynomial
$p \in L_n$
we have
$S^{+}(p)\le n-1$
.
Historical comments. S. Karlin and W.J. Studden [Reference Karlin and Studden21, Ch. VII] introduced
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-systems via condition (c) and showed that (c) yields (a). In the matrix form the implication
$\text {(b)}\Leftrightarrow \text {(c)}$
can be found in the recent paper [Reference Morozov, Zheltkov and Osinsky31]. Corollary 1.5 was given in [Reference Karlin and Studden21, Ch. VII, § 1] in the particular case when T-systems are generated by strictly total positive kernels. Also, Corollary 1.6 was proved in [Reference Karlin and Studden21, Ch. VII, § 2] using property (c) of Theorem 1.4. We will give a different proof of Corollary 1.6 based on an oscillation property of discrete polynomials (2.4).
We note that the discrete Chebyshev approximation plays an important role in numerical analysis (see, e.g., [Reference Dunham11] and [Reference Morozov, Zheltkov and Osinsky31, Reference Zamarashkin, Morozov and Tyrtyshnikov38] for the best approximation by low-rank matrices). In these problems the use of effective algorithms such as the Remez algorithm is crucial. Let us mention that versions of Theorem 1.3 also allow one to construct effective algorithms for non-Chebyshev systems, see [Reference Protasov and Kamalov32]. Moreover, C. Dunham [Reference Dunham11] developed a discrete Remez algorithm based on alternance sets. Thus, Theorem 1.4 completes this study as it characterizes all systems that admit alternation.
For applications, it is important to construct various discrete Chebyshev systems. To this end, in the next section we consider the eigenfunctions of Sturm-Liouville problems.
1.2 Sturm oscillation theorem, T-systems, and orthogonal polynomials
We start with the following known fact: T-systems can be obtained using Sturm-Liouville problems. In 1836, C. Sturm proved the following remarkable result, which was obtained in the same year by J. Liouville using a different method under less restrictions. See [Reference Bérard and Helffer5, Reference Simon33, Reference Steinerberger34] for the historical comments.
Sturm’s theorem Let
$\{V_{k}\}_{k=1}^{\infty }\subset C^{2}[a,b]$
be the set of eigenfunctions associated to eigenvalues
$\rho _{1}<\rho _{2}<\dots $
of the following Sturm-Liouville problem:
$$ \begin{align} \begin{gathered} (K(t)V'(t))'+(\rho G(t)-L(t))V(t)=0,\quad t\in [a,b],\\ (KV'-hV)(a)=0,\quad (KV'+HV)(b)=0, \end{gathered} \end{align} $$
where
$G,K,L\in C[a,b]$
,
$K\in C^{1}(a,b)$
,
$K,G>0$
on
$(a,b)$
,
$h,H\in [0,\infty ]$
, and
$\rho $
denotes the spectral parameter. Then
-
(a) Every k-th eigenfunction
$V_{k}$
has exactly
$k-1$
simple zeros in
$(a,b)$
. -
(b) For any nontrivial real polynomial of the form
(1.6)the number of distinct zeros of P on
$$ \begin{align} P(t)=\sum_{k=m}^{n}A_{k}V_{k}(t),\quad m,n\in \mathbb{N},\quad m\le n, \end{align} $$
$(a,b)$
is at least
$m-1$
and at most
$n-1$
.
Remark 1.7. (i) Part (a) is sometimes called the weak Sturm oscillation theorem [Reference Steinerberger34] and proved using comparison theorems (see, e.g., [Reference Levitan and Sargsjan26] and [Reference Simon33]). Part (b) is a much stronger result (the Sturm oscillation theorem). As noted by P. Bérard and B. Helffer [Reference Bérard and Helffer5], while part (a) is a well-known statement, part (b) has been almost forgotten. However, for the trigonometric system, it has been known since the early 20th century as the Sturm-Hurwitz theorem: If a real Fourier series has a spectral gap, that is,
$f(t) = \sum _{k=m}^\infty (a_k \cos kt + b_k \sin kt)$
, then f has at least
$2m$
sign changes over the period. Sturm proved this fact for polynomials (cf. (1.6)), while Hurwitz extended it to the general case. Similar results on spectral gap problems are closely related to the Fourier uncertainty principle (see, e.g., [Reference Eremenko and Novikov14, Reference Gorbachev, Ivanov and Tikhonov17, Reference Gorbachev, Ivanov and Tikhonov18, Reference Logan27, Reference Mitkovski and Poltoratski29, Reference Steinerberger34, Reference Ulanovskii37]) and signal processing (see, e.g., [Reference Mao28]).
(ii) It follows from Sturm’s theorem that
$\{V_{k}(t)\}_{k=1}^{n}$
forms a T-system on the interval
$(a,b)$
. By Corollary 1.5, for
$a<0$
and
$b>q$
, the set
$\{V_{k}(\nu )\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system.
Our main goal in this section is to obtain the Sturm oscillation theorem for eigenfunctions of a discrete Sturm-Liouville problem on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. This problem is treated in detail in several monographs, for example, [Reference Agarwal1, Ch. 12], [Reference Atkinson3, Ch. 4], [Reference Kelley and Peterson22, Ch. 7]. Moreover, discrete comparison and separation theorems in various settings have been studied in, for example, [Reference Agarwal, Bohner, Grace and O’Regan2, Reference Elaydi13, Reference Simon33, Reference Teschl36]. However, an explicit discrete analogue of Sturm’s theorem has been an open question.
To formulate the discrete Sturm-Liouville problem, we use infinite tridiagonal Jacobi matrix, which is, in general, nonsymmetric. Let
$\{\alpha _{l}\}_{l=0}^{\infty }$
be a sequence of real numbers,
$\{\gamma _{l}\}_{l=0}^{\infty }$
,
$\{\beta _{l}\}_{l=0}^{\infty }$
,
$\{\rho _l\}_{l=0}^{\infty }$
be sequences of positive numbers. Define
Considering the eigenvalue problem
$$ \begin{align*}JP=\lambda \rho P\end{align*} $$
with an infinite vector
$P=\{P_l(\lambda )\}_{l=0}^{\infty }$
and a diagonal matrix
$\rho =\operatorname {diag}(\{\rho _l\}_{l=0}^{\infty })$
, we arrive at the Sturm-Liouville problem in the recurrence relation form:
$$ \begin{align} \begin{gathered} \gamma_{l-1}P_{l-1}(\lambda)+\alpha_lP_l(\lambda)+\beta_{l}P_{l+1}(\lambda)=\lambda \rho_{l}P_l(\lambda),\quad l\in \mathbb{Z}_{+},\quad \lambda\in\mathbb{C},\\ P_{-1}(\lambda)=0,\quad P_0(\lambda)=1, \end{gathered} \end{align} $$
where
$\gamma _{-1}>0$
is arbitrary. It is clear that the solution of this problem is given by a family of algebraic polynomials
$P_l(\lambda )$
of degree l with positive leading coefficients.
For a nondecreasing function
$\mu (\lambda )$
on
$\mathbb {R}$
, the set
$$\begin{align*}\mathcal{S}(\mu)=\{\lambda\colon \mu(\lambda+\delta)-\mu(\lambda-\delta)> 0\ \text{for all}\ \delta>0\} \end{align*}$$
is called the spectrum of
$\mu $
. To attack Problem (1.7), we apply the Favard theorem (see [Reference Chihara8, Th. II. 6.4]), which leads to the following statement.
Theorem 1.8.
-
(a) Polynomials
$P_l(\lambda )$
are orthogonal with respect to a positive measure
$\mu $
on
$\mathbb {R}$
, defined by a nondecreasing function of bounded variation with infinite spectrum, and (1.8)
$$ \begin{align} d_{l}\int_{\mathbb{R}}P_l(\lambda)P_m(\lambda)\,d\mu(\lambda)=\delta_{lm},\quad d_0=\rho_0,\quad d_l=\rho_{l}\,\frac{\beta_0\cdots \beta_{l-1}}{\gamma_0\cdots \gamma_{l-1}},\quad l\ge 1. \end{align} $$
-
(b) A measure
$\mu $
has a finite support
$[a,b]$
, the endpoints of which will be the limit points of zeros of the orthogonal polynomials
$P_l(\lambda )$
, if and only if the sequences (1.9)are bounded.
$$ \begin{align} \frac{\alpha_{l}}{\rho_{l}},\quad \frac{\gamma_{l-1}\beta_{l-1}}{\rho_{l-1}\rho_{l}} \end{align} $$
Note that positivity of coefficients
$\gamma _l$
,
$\beta _l$
,
$\rho _l$
is a natural condition in Theorem 1.8, see Theorem II.6.4 in [Reference Chihara8].
Throughout the paper we assume that
$ q\in \mathbb {Z}_+$
and
$\eta \in \mathbb {R}.$
Consider the polynomial
$$ \begin{align} \widetilde{P}_{q+1}(\lambda)=P_{q+1}(\lambda)-\eta P_{q}(\lambda) \end{align} $$
and define its zeros by
$ \lambda _{q+1}<\dots <\lambda _{1}, $
see [Reference Szegö35, Ch. III, § 3.3]. We also define the following set of discrete functions:
$$ \begin{align} \psi_k(\nu)=P_{\nu}(\lambda_k),\quad k\in [1,q+1]_{{\scriptscriptstyle\mathbb{Z}}},\quad \nu\in [0,q]_{{\scriptscriptstyle\mathbb{Z}}}. \end{align} $$
By virtue of (1.7),
$\lambda _k$
and
$\psi _k(\nu )$
are eigenvalues and eigenfunctions of the discrete Sturm-Liouville problem
$$ \begin{align} \begin{gathered} \gamma_{\nu-1}\psi(\nu-1)+\alpha_{\nu}\psi(\nu)+\beta_{\nu}\psi(\nu+1)=\lambda \rho_{\nu}\psi(\nu),\quad \nu\in [0,q]_{{\scriptscriptstyle\mathbb{Z}}},\\ \psi(-1)=0,\quad \psi(0)=1,\quad \psi(q+1)=\eta\psi(q). \end{gathered} \end{align} $$
Introducing the Jacobi matrix
problem (1.12) corresponds to the eigenvalue problem
$$ \begin{align*}J_{q+1}\psi=\lambda \rho \psi.\end{align*} $$
Sturm’s theory on the zeros of discrete polynomials with respect to the set
$\{\psi _k(\nu )\}_{k=1}^{q+1}$
(see (1.11)) is similar to that in the continuous case (see (1.5)), with the additional use of the quantities
$S^{-}(f)$
and
$S^{+}(f)$
.
Before stating an analogue of item (a) of Sturm’s theorem given in Theorem 1.10 below, we mention the following result by B. Simon.Footnote 4
Theorem 1.9 [Reference Simon33, Th. 2.3]
For
$n\in [1,q]_{\scriptscriptstyle \mathbb {Z}},$
there holds
$$\begin{align*}|\{j\in [1,n]_{\scriptscriptstyle\mathbb{Z}}\colon \lambda<\lambda_{j,n}\}|= |\{\nu\in [1,n]_{\scriptscriptstyle\mathbb{Z}}\colon \operatorname{sign} P_{\nu-1}(\lambda)\ne \operatorname{sign} P_{\nu}(\lambda)\}|, \end{align*}$$
where
$\lambda _{l,l}<\dots <\lambda _{1,l}$
are the zeros of the polynomials
$P_l(\lambda )$
,
$\lambda \ne \lambda _{j,l}$
for
$1\le j\le l\le n$
, and
$|I|$
denotes the cardinality of a finite set I.
Taking into account that the function
$\psi _k$
cannot have two consecutive zeros of the first type, this theorem implies the following result.
Theorem 1.10. For all
$k\in [1,q+1]_{{\scriptscriptstyle \mathbb {Z}}},$
$$\begin{align*}S^{-}(\psi_k)=S^{+}(\psi_k)=N(\psi_k)=k-1. \end{align*}$$
Note that Theorem 1.10 also follows from Theorem 1 in [Reference Gantmacher and Krein15, Ch. II] whose proof is based on the use of the concepts of lines and nodes for the piecewise linear function associated with Jacobi matrices (see Definition 2 in [Reference Gantmacher and Krein15, Ch. II]). Moreover, the proof of the fact that
$N(\psi _k)=k-1$
based on the discrete Sturm separation theorem can be found in [Reference Kelley and Peterson22, Theorem 7.6]. The corresponding results for self-adjoint systems are given in [Reference Došlý and Kratz9, Theorem 2], [Reference Došlý, Elyseeva and Hilscher10, Corollary 5.79], [Reference Kratz23, Theorem 1]. In Section 3, we present a simple proof of Theorem 1.10 based on the property of the interlacing of zeros of orthogonal polynomials (see (3.2)).
Our main result in this section is the discrete counterpart of item (b) of Sturm’s theorem, cf. Remark 1.7.
Theorem 1.11. Let integers m and n satisfy
$1\le m\le n\le q+1$
. For any nontrivial polynomial
$$ \begin{align} V(\nu)=\sum_{k=m}^{n}a_k\psi_k(\nu), \end{align} $$
we have
$$ \begin{align} m-1\le S^{-}(V)\le N(V)\le S^{+}(V)\le n-1. \end{align} $$
The proof of Theorem 1.11 is based on Theorem 1.10 and a generalization of Liouville’s method (see [Reference Bérard and Helffer5]). Note also that the inequalities
$m-1\le S^{-}(V)$
and
$S^{+}(V)\le n-1$
for
$\eta =0$
and
$\rho _l=1$
were proven in [Reference Gantmacher and Krein15, Ch. II, Th. 6] under the assumption that the Jacobi matrix
$J_{q+1}$
is oscillatory. Recall that a square matrix is called oscillatory if it is totally non-negative, and some power of it is totally positive (see [Reference Gantmacher and Krein15, Ch. II, Definition 4]). In our case, in general, the matrix
$J_{q+1}$
is not oscillatory and it becomes oscillatory only if all its principal minors are positive [Reference Gantmacher and Krein15, Ch. II, Th. 11]. It is worth mentioning that the estimate
$N(V)\le n-1$
was mentioned in [Reference Atkinson3, Problem 2, p. 539].
Corollary 1.12. For any
$n\in [1,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
, the set
$\{\psi _k(\nu )\}_{k=1}^{n}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system.
To conclude, we state the discrete spectral gap theorem (cf. Remark 1.7).
Corollary 1.13. Let
$m\in [1,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
be such that
$f(\nu )=\sum _{k=m}^{q+1}a_k\psi _k(\nu )$
. Then
$m-1\le S^{-}(f)\le N(f)$
.
1.3 Monotonicity property of coefficients of polynomials with removed largest zeros
Consider polynomial (1.10) and divide it into
$\prod _{j=1}^{m+1} (\lambda -\lambda _{j})$
with
$m+1$
largest zeros
$\lambda _{j}$
,
$m\le q$
. Taking into account (1.8), we expand it into the Fourier sum
$$ \begin{align} \frac{\widetilde{P}_{q+1}(\lambda)}{(\lambda-\lambda_{1})\cdots (\lambda-\lambda_{m+1})}= \sum_{l=0}^{q-m}d_{l}a_{l}P_{l}(\lambda). \end{align} $$
The question arises whether all coefficients
$a_{l}\ge 0$
. It turns out that the answer is affirmative. The case
$m=0$
is simple and corresponds to the Christoffel-Darboux formula, while the case
$m=1$
was proven in [Reference Gorbachev and Ivanov16]. In full generality, this nontrivial fact was proven by H. Cohn and A. Kumar [Reference Cohn and Kumar7, Th. 3.1] while working on the problem on discrete energy on the Euclidean sphere.
In this section, using the discrete Sturm theorem, we take one step further and establish the monotonicity property of the Fourier coefficients for all m, which also implies that
$a_{l}\ge 0$
. This fact is crucial to deal with a spectral gap problem in Section 1.4.
Also, we note that the related polynomial
$p_m(\lambda )=\frac {P_{q+1}^2(\lambda )}{(\lambda -\lambda _{1})\cdots (\lambda -\lambda _{m+1})}$
appeared in various extremal problems. For example, S. Bernstein used
$p_0$
to disprove the existence of Chebyshev’s quadrature formula. Moreover, this polynomial is an extremizer in the following question (see [Reference Babenko4, Reference Montgomery and Ulrike30]): find
$$ \begin{align*}\inf_{a_k, b_k} \operatorname{mes}\,\Big\{x\in (-\pi,\pi]\colon \sum_{k=1}^n (a_k\cos kx+b_k\sin kx)>0\Big\}.\end{align*} $$
V. Yudin used
$p_m$
in problems of multiple covering of the torus [Reference Yudin40] and the Euclidean sphere [Reference Yudin41].
Our main result here is the following theorem.
Theorem 1.14. Let
$q\in \mathbb {N}$
and
$m\in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. Let also
$(a,b)$
be an interval containing all the zeros of the polynomial
$P_{q }(\lambda )$
and
$\eta _{b}=P_{q+1}(b)/P_{q}(b)$
.
The following inequalities hold for the coefficients
$a_l$
in the expansion (1.15)
$:$
-
(a) if either
$m\ge 1$
and
$\eta \in \mathbb {R}$
or
$m=0$
and
$\eta <\eta _{b}$
, then (1.16)
$$ \begin{align} \frac{a_0}{P_0(b)}>\frac{a_1}{P_1(b)}>\cdots>\frac{a_{q-m}}{P_{q-m}(b)}>0; \end{align} $$
-
(b) if
$m=0$
and
$\eta =\eta _{b}$
, then (1.17)
$$ \begin{align} \frac{a_0}{P_0(b)}=\frac{a_1}{P_1(b)}=\cdots=\frac{a_{q}}{P_{q}(b)}>0; \end{align} $$
-
(c) if
$m=0$
and
$\eta>\eta _b$
, then (1.18)
$$ \begin{align} 0<\frac{a_0}{P_0(b)}<\frac{a_1}{P_1(b)}<\cdots<\frac{a_{q}}{P_{q}(b)}. \end{align} $$
For polynomials normalized by
$P_{l}(1)=1$
, this result has a simpler form.
Corollary 1.15. Let
$P_{l}(\lambda )$
be orthogonal on
$[-1,1]$
and
$P_{l}(1)=1$
. Then
$$\begin{align*}{\begin{cases} a_0>a_1>\cdots>a_{q-m}>0,&\text{if either } m=0 \text{ and } \eta<1 \text{ or } m\ge1 \text{ and } \eta\in\mathbb{R}, \\ a_0=a_1=\cdots=a_{q}>0,&\text{if } m=0 \text{ and } \eta=1,\\ 0<a_0<a_1<\cdots<a_{q},&\text{if } m=0 \text{ and } \eta>1. \end{cases}} \end{align*}$$
1.4 An extremal problem for polynomials with spectral gap
Let
$m,n\in \mathbb {Z}_+$
,
$m<n$
. Let also
$\mu $
be the measure on
$[-1,1]$
from Theorem 1.8 and
$\{U_{l}\}$
be the set of polynomials orthogonal on
$[-1,1]$
with respect to
$d\mu $
with normalization
$U_{l}(1 )=1$
,
$d_l\int _{-1}^1U_lU_m\,d\mu =\delta _{lm}$
.
We consider the following problem: find
$\lambda \in [-1,1]$
and an algebraic polynomial of degree n on
$[-1,1]$
with non-negative Fourier coefficients in
$\{U_{l}\}$
having a spectral gap of the length m (equivalently, with the first m moments being zero), which preserves its sign over the largest possible interval
$[-1,\lambda ]$
.
Let
$\Pi $
be the set of real algebraic polynomials and
$\Pi _n$
consist of those of degree at most n. By
$\Pi _{+}(\{U_{l}\})$
we denote a subset of
$p\in \Pi $
such that all coefficients
$a_{l}$
in the Fourier expansion
$$ \begin{align*} p(t)=\sum_{l}d_{l}a_{l}U_{l}(t),\quad a_{l}=a_l(p)=\int_{-1}^{1}p(t)U_{l}(t)\,d\mu(t), \end{align*} $$
are non-negative.
Set
$$ \begin{align*} K_{n}(\mu,m)=\Big\{p\in\Pi_n\cap\Pi_{+}(\{U_l\})\colon \mu_i(p)=0,\ i=0,\dots,m-1,\ \mu_{m}(p)\ge 0\Big\}, \end{align*} $$
where
$\mu _i(p)=\int _{-1}^{1}t^ip(t)\,d\mu (t)$
is the ith moment of the polynomial p. Note that the conditions
$\mu _i(p)=0$
are equivalent to the fact that
$a_i(p)=0$
,
$ i=0,\dots ,m-1.$
Our goal is to find the quantity
$$\begin{align*}B_n(\mu,m)=\sup_{p\in K_{n}(\mu,m)\setminus \{0\}}\{\lambda\in [-1,1]\colon (-1)^{m-1}p(t)\ge 0,\ -1\le t\le\lambda\}. \end{align*}$$
This question turns out to play a crucial role in the study of spherical codes and designs. For
$m=1$
it was posed by V. Yudin [Reference Yudin39]. Specifically, for
$m=1$
and
$n=2q$
, he found extremizers for this problem, though the positivity of the Fourier coefficients was not proven. Further results in this direction can be found in [Reference Gorbachev and Ivanov16, Reference Gorbachev, Ivanov and Tikhonov17, Reference Gorbachev, Ivanov and Tikhonov18]; see also [Reference Cohn and Kumar7, Reference Levenshtein25].
Let
$\{U_l^{(1)}(t)\}_{l=0}^{\infty }$
be the set of polynomials with normalization
$U_l^{(1)}(1) =1$
, orthogonal on
$[-1,1]$
with respect to the measure
$d\mu ^{(1)}(t)=(1+t)\,d\mu (t)$
. The zeros of the polynomials
$U_{q+1}(t)$
and
$U_{q+1}^{(1)}(t)$
, numbered in the descending order, are denoted by
$t_k$
and
$t_k^{(1)}$
,
$k=1,\dots ,q+1$
, respectively.
We say that the set of orthogonal polynomials
$\{V_l\}$
satisfies the Krein property if
$V_mV_n\in \Pi _{+}(\{V_l\})$
, that is,
$V_{m}V_{n}=\sum _{k=|m-n|}^{m+n}c_{m,n,k}V_{k}$
with
$c_{m,n,k}\ge 0$
, for any
$m,n\in \mathbb {Z}_+$
, see [Reference Levenshtein25].
Theorem 1.16. Let
$m\le q$
and the set
$\{U_l\}$
satisfy the Krein property.
-
(a) If
$n=2q-m+1$
, then (1.19)and the unique extremal polynomial has the form (up to a positive constant)
$$ \begin{align} B_n(\mu,m)=t_{m+1} \end{align} $$
(1.20)
$$ \begin{align} p_{n,1}^{*}(t)=\frac{U_{q+1}^2(t)}{(t-t_{1})\cdots(t-t_{m+1})}. \end{align} $$
-
(b) If
$n=2q-m+2$
and either the set
$\{U_l^{(1)}\}$
satisfies the Krein property or
$\mu (t)$
is an odd function, then (1.21)and the unique extremal polynomial has the form (up to a positive constant)
$$ \begin{align} B_n(\mu,m)=t_{m+1}^{(1)} \end{align} $$
(1.22)
$$ \begin{align} p_{n,2}^{*}(t)=\frac{(1+t)(U_{q+1}^{(1)}(t))^2}{(t-t_{1}^{(1)})\cdots(t-t_{m+1}^{(1)})}. \end{align} $$
Moreover, we have
$\mu _{m}(p_{n,j}^{*})=a_m(p_{n,j}^{*})=0$
,
$j=1,2$
.
We note that a version of Yudin-type problem for a wider class of polynomials p (without the condition
$p\in \Pi _{+}(\{U_l\})$
) was solved in [Reference Ivanov20]. Our proof of Theorem 1.16 is based on Corollary 1.15.
1.5 Structure of the paper
The rest of the paper is organized as follows. In Section 2, we provide the proofs of Chebyshev’s theorem for
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-systems (Theorem 1.4) and the important Corollary 1.6, which claims that
$S^{+}(p)\le n-1$
for a polynomial
$p \in L_n$
, where
$p(\nu )=\sum _{k=1}^{n}a_k\varphi _k(\nu )$
and
$\{\varphi _k\}_{k=1}^n\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system.
In Section 3, we study the discrete Sturm oscillation theory and its connection with T-systems and prove Theorems 1.8, 1.10, and 1.11.
Section 4 is devoted to the proof of Theorem 1.14. In Section 5, we study polynomials with spectral gaps and solve Yudin’s extremal problem (Theorem 1.16).
In the Appendix, we explicitly compute the determinants (4.1) for the Jacobi polynomials corresponding to the classical trigonometric systems. In particular, we verify that, for all
$\nu \in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-systems,
$$ \begin{align*}\operatorname{sign} \Delta\left( \begin{matrix} \varphi_1, & \dots, &\varphi_{m},& \varphi_{m+1}\\ \nu_{m}, & \dots, &\nu_1, &\nu \end{matrix}\right) =\operatorname{sign} \Delta\left( \begin{matrix} \varphi_1, & \dots, &\varphi_{m},& \varphi_{m+1}\\ q, & \dots, &q-m+1, &0 \end{matrix}\right) \operatorname{sign} \prod_{j=1}^{m}(\nu_j-\nu), \end{align*} $$
which in the general case will be shown in Lemma 4.1.
2 Proofs of Theorem 1.4 and Corollary 1.6
Recall that
$q\in \mathbb {Z}_+$
and
$\eta \in \mathbb {R}$
. The proof of Theorem 1.4 is based on Lemmas 2.1–2.5 below. First, in addition to the characteristic property of
$T_0$
-systems given by Proposition 1.2, we present another necessary and sufficient condition.
Lemma 2.1 [Reference Laurent24, Proposition 3.4.2]
A set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_0$
-system if and only if, for any integers
$0\le \nu _1<\dots <\nu _n\le q$
and any points
$\{y_i\}_{i=1}^n\subset \mathbb {R}$
, there is a unique polynomial (1.3), for which
$$\begin{align*}p(\nu_i)=y_i,\quad i=1,\dots,n. \end{align*}$$
Second, we assume that the set
$\{\varphi _k\}_{k=1}^{n} \subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_0$
-system and the determinants (1.4) are nonzero for any integers
$0 \leq \nu _1 < \dots < \nu _n \leq q$
.
Set, for
$0\le \nu _1<\dots <\nu _{n+1}\le q$
,
$$ \begin{align*} \Delta_{\nu_1}&=\Delta\left(\begin{matrix} \varphi_1,&\dots,&\varphi_n\\ \nu_2,&\dots,&\nu_{n+1} \end{matrix}\right),\quad \Delta_{\nu_{n+1}}=\Delta\left( \begin{matrix} \varphi_1,&\dots,&\varphi_n\\ \nu_1,&\dots,&\nu_{n} \end{matrix}\right), \\ \Delta_{\nu_i}&=\Delta\left( \begin{matrix} \varphi_1,&\dots,& \varphi_{i-1},& \varphi_{i},&\dots,& \varphi_n\\ \nu_1,&\dots,&\nu_{i-1},&\nu_{i+1},&\dots,& \nu_{n+1} \end{matrix}\right), \quad i=2,\dots,n. \end{align*} $$
Lemma 2.2. Let
$0\le \nu _1<\dots <\nu _{n+1}\le q$
. The linear functional
$$ \begin{align} l(p)=\sum_{i=1}^{n+1}\gamma_{i}p(\nu_i)=0 \end{align} $$
for any polynomial
$p\in L_n$
if and only if, up to a nonzero constant,
$\gamma _{i}=(-1)^i\Delta _{\nu _i}$
,
$i=1,\dots ,n+1$
.
Proof. First, we note that
$p\in L_n$
only if
$$\begin{align*}\Delta\left( \begin{matrix} p,&\varphi_1, & \varphi_2, & \dots, & \varphi_n\\ \nu_1,&\nu_2, & \nu_3, & \dots, & \nu_{n+1} \end{matrix}\right) =0. \end{align*}$$
Expanding the determinant along the first column, we obtain
$$ \begin{align} \sum_{i=1}^{n+1}(-1)^{i}\Delta_{\nu_i}p(\nu_i)=0. \end{align} $$
On the other hand, from (2.1), for any
$a_1,\dots ,a_k$
,
$$\begin{align*}0=\sum_{i=1}^{n+1}\gamma_i\sum_{k=1}^{n}a_k\varphi_k(\nu_i)=\sum_{k=1}^{n}a_k\sum_{i=1}^{n+1}\gamma_i\varphi_k(\nu_i), \end{align*}$$
and therefore,
$$ \begin{align} \sum_{i=1}^{n+1}\gamma_i\varphi_k(\nu_i)=0,\quad k=1,\dots,n. \end{align} $$
Since
$\Delta _{\nu _i}\neq 0$
, all the solutions
$(\gamma _1,\dots ,\gamma _{n+1})$
of the system (2.3) form a one-dimensional space. Taking into account (2.2), we arrive at the statement of the lemma.
Lemma 2.3. Let the set
$\{\varphi _k\}_{k=1}^{n}\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
be a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system. The polynomial
$p^{*}\in L_n$
is the best uniform approximant of a function
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
if and only if
$p^{*}$
admits an alternance set of length
$n+1$
.
Proof. The sufficiency follows from Theorem 1.3. To prove the necessity, according to Lemma 2.2 and Theorem 1.3, we need to show that for a linear functional
$l(p)$
on
$L_n$
satisfying property (2.1), there holds
$\gamma _i\gamma _{i+1}<0$
for
$i=1,\dots ,n$
. We will follow the reasoning given in the proof of Proposition 3.5.1 in [Reference Laurent24]. Applying Lemma 2.1, we construct polynomials
$p_i\in L_n$
satisfying
$$\begin{align*}p_i(\nu_j)=0,\quad j=1,\dots,i-1, i+2,\dots,n+1,\quad p_i(\nu_i)=1. \end{align*}$$
Since
$p_i$
has
$n-1$
zeros of the first type, it follows from the definition of the
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system that
$p_i$
has no zeros of the second type and, therefore,
$p_i( \nu _{i+1})>0$
. From this,
$$\begin{align*}l(p_i)=\gamma_ip_i(\nu_i)+\gamma_{i+1}p_{i}(\nu_{i+1})=0\quad \text{and}\quad \gamma_i\gamma_{i+1}<0, \end{align*}$$
completing the proof.
Lemma 2.4.
(a) For any
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, its best uniform approximant
$p^*\in L_n$
admits an alternance set of length
$n+1$
if and only if all determinants (1.4), constructed over all sequences
$0\le \nu _1<\dots <\nu _{n}\le q$
, have the same sign.
(b) If all determinants (1.4) have the same sign, then any sequence
$0\le \nu _1<\dots <\nu _{n+1}\le q$
is an alternance set of length
$n+1$
for the polynomial
$p^*\in L_n$
, which is the best approximant of some function
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
.
Proof. Part (a) follows from Theorem 1.3 and Lemma 2.2.
To obtain part (b), the sequence
$0\le \nu _1<\dots <\nu _{n+1}\le q$
is an alternance set of length
$n+1$
, for example, for the function
$$\begin{align*}f(\nu)= \begin{cases} (-1)^{i-1},&\nu=\nu_{i},\ i=1,\dots,n+1,\\ 0&\text{otherwise}. \end{cases} \end{align*}$$
Note that, by virtue of Theorem 1.3, the best uniform approximant is
$p^*(\nu )\equiv 0$
, and
$E(f,L_n)_{\infty }=1$
. The proof is now complete.
Lemma 2.5. Let
$p\in L_n$
and
$N(p)=n$
. There exists a collection of points
$\Omega =\{0\le \nu _1<\dots <\nu _{n+1}\le q\}$
such that if for some positive sequence
$\{\rho _i\}_{i=1}^{n+1}$
there holds
$$ \begin{align} \sum_{i=1}^{n+1}(-1)^{i}\rho_ip(\nu_i)=0, \end{align} $$
then
$p\equiv 0$
.
Proof. Without loss of generality, we can assume that all
$\rho _i=1$
. If the polynomial p has n zeros of the first type, then by the definition of
$T_0$
-systems,
$p\equiv 0$
.
Let p have a sign change. An interval
$[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}}\subset [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$l\ge 1$
, is called a closed segment of sign changes if
$p(\nu -1) p(\nu )<0$
for
$\nu =k+1,\dots ,k+l$
and
$p(k-1)p(k)\ge 0$
,
$p(k+l)p( k+l+1)\ge 0$
. It consists of
$l+1$
points and contains l zeros of the second type.
Let the polynomial p have t zeros of the first type and s closed segments of sign changes
$[k_j,k_j+l_j]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$j=1,\dots ,s$
, where
$k_{j+1}-k_j-l_j\ge 1$
. Then
$l_1+\dots +l_s=n-t$
. We form the set
$\Omega $
as follows:
$\Omega $
consists of all t zeros of the first type and
$n+1-t$
points from the closed segments
$\Omega _j$
,
$j=1,\dots ,s$
, of sign changes. In more detail, set
$\Omega _1=[k_1,k_1+l_1]_{{\scriptscriptstyle \mathbb {Z}}}$
and construct
$\Omega _j$
,
$j=2,\dots ,s$
, such that either
$\Omega _j=[k_j,k_j+l_j-1]_{{\scriptscriptstyle \mathbb {Z}}}$
or
$\Omega _j=[k_j+1 ,k_j+l_j]_{{\scriptscriptstyle \mathbb {Z}}}$
. Note that the number of points in closed segments is equal to
$(l_1+1)+l_2+\dots +l_s=n+1-t$
.
If
$r_1+i$
,
$i=0,\dots ,l_1$
, are the indices of the points
$k_1+i$
of the set
$\Omega $
in
$\Omega _1$
and
$p(k_1+i)=(-1)^{i_1 +k_1+i}|p(k_1+i)|$
, then
$k_1+i=\nu _{r_1+i}$
and
$$\begin{align*}\sum_{i=0}^{l_1}(-1)^{r_1+i}p(\nu_{r_1+i})=(-1)^{i_1+k_1+r_1}\sum_{i=0}^{l_1}|p(\nu_{r_1+i})|. \end{align*}$$
Let the segments
$\Omega _m$
, for
$m=1,\dots ,j-1$
, have already been constructed. Let
$r_j+i$
,
$i=0,\dots ,l_j-1$
, be the indices of points of the set
$\Omega $
in
$\Omega _j$
and
$p(k_j+i)=(-1)^{i_j+k_j+i}|p(k_j+i)|$
,
$i=0,\dots ,l_j$
.
If
$(-1)^{i_j+k_j+r_j}=(-1)^{i_1+k_1+r_1}$
, then we set
$\nu _{r_j+i}=k_j+i$
,
$i=0, \dots ,l_j-1$
. We have
$$\begin{align*}\sum_{i=0}^{l_j-1}(-1)^{r_j+i}p(\nu_{r_j+i})=\sum_{i=0}^{l_j-1}(-1)^{r_j+i}p(k_j+i)=(-1)^{i_1+k_1+r_1}\sum_{i=0}^{l_j-1}|p(\nu_{r_j+i})|. \end{align*}$$
If
$(-1)^{i_j+k_j+r_j+1}=(-1)^{i_1+k_1+r_1}$
, then we put
$\nu _{r_j+i}=k_j+i+1$
,
$ i=0,\dots ,l_j-1$
. Again, we have
$$\begin{align*}\sum_{i=0}^{l_j-1}(-1)^{r_j+i}p(\nu_{r_j+i})=\sum_{i=0}^{l_j-1}(-1)^{r_j+i}p(k_j+i+1)=(-1)^{i_1+k_1+r_1}\sum_{i=0}^{l_j-1}|p(\nu_{r_j+i})|. \end{align*}$$
Thus, the set
$\Omega =\{0\le \nu _1<\dots <\nu _{n+1}\le q\}$
is constructed. For this set, in view of (2.4), there holds
$$ \begin{align*} 0&=\sum_{i=1}^{n+1}(-1)^{i}p(\nu_i)=\sum_{i=0}^{l_1}(-1)^{r_1+i}p(\nu_{r_1+i})+\sum_{j=2}^s\sum_{i=0}^{l_j-1}(-1)^{r_j+i}p(\nu_{r_j+i})\\& =(-1)^{i_1+k_1+r_1}\Big\{\sum_{i=0}^{l_1}|p(\nu_{r_1+i})|+\sum_{j=2}^s\sum_{i=0}^{l_j-1}|p(\nu_{r_j+i})|\Big\}. \end{align*} $$
Hence,
$p(\nu _i)=0$
for
$i=1,\dots ,n+1$
. Then we deduce that
$p\equiv 0$
, completing the proof.
Proof of Theorem 1.4
By Lemma 2.3, (a) implies (b) and by Lemma 2.4, items (b) and (c) are equivalent. Finally, applying Lemmas 2.2 and 2.5, (a) follows from (b). The proof is now complete.
Proof of Corollary 1.6
To prove this result, it is sufficient to show the following analogue of Lemma 2.5:
If
$S^{+}(p)=n$
, then there exists a set
$\Omega =\{0\le \nu _1<\dots <\nu _{n+1}\le q\}$
such that if for some positive numbers
$\{\rho _i\}_{i=1}^{n+1}$
property (2.4) holds, then
$p\equiv 0$
.
The construction of the set
$\Omega $
is similar to the one in Lemma 2.5, so we will only sketch the proof. As above, we assume that each
$\rho _i=1$
.
Let us consider four different types of intervals. An interval of the first type is a closed segment of sign changes (see the proof of Lemma 2.5). An interval of the second type is
$I=[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}}$
, for which
$$\begin{align*}p(k)p(k+l)>0,\quad p(k+1)=\dots=p(k+l-1)=0\quad \text{and}\quad l-1 \text{ is odd}. \end{align*}$$
An interval of the third type is
$I=[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}}$
, for which
$$\begin{align*}p(k)p(k+l)<0,\quad p(k+1)=\dots=p(k+l-1)=0\quad \text{and}\quad l-1 \text{ is even}. \end{align*}$$
It is worth mentioning that intervals of the second and third types contribute to an increase in the number of zeros of the polynomial when zero values are changed to nonzero values, as
$N(p,[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}})=l-1$
and
$S^{+}(p,[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}})=l$
.
Intervals of the fourth type are
$I=[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}}$
, consisting of zeros of the first type, for which one of the following conditions holds: (i)
$k=0$
, (ii)
$k+l=q$
, (iii)
$l+1$
is odd and
$p(k-1)p(k+l+1)<0$
, (iv)
$l+1$
is even and
$p(k-1)p(k +l+1)>0$
. Note that changing zero values of p to any nonzero numbers at points of intervals of the fourth type cannot increase the number of its zeros.
Denote the sets of intervals of the first, second, third, and fourth types by
$\Sigma _1$
,
$\Sigma _2$
,
$\Sigma _3$
, and
$\Sigma _4$
, respectively. Set
$\Sigma =\cup _{i=1}^{3}\Sigma _i$
. Note that intervals from different sets
$\Sigma _i$
can intersect only at their endpoints.
We have
$$\begin{align*}n=\sum_{I\in\Sigma_1}(|I|-1)+\sum_{I\in\Sigma_2}(|I|-1)+\sum_{I\in\Sigma_3}(|I|-1)+\sum_{I\in\Sigma_4}|I|. \end{align*}$$
By
$\Omega $
we define a set of allFootnote
5
points of the intervals from
$\Sigma $
and
$\Sigma _4$
. Then the cardinality of
$\Omega $
is
$$\begin{align*}1+\sum_{I\in\Sigma_1}(|I|-1)+\sum_{I\in\Sigma_2}(|I|-1)+\sum_{I\in\Sigma_3}(|I|-1)+\sum_{I\in\Sigma_4}|I|=n+1. \end{align*}$$
Let now
$[\nu _{r_1},\nu _{r_1}+l_{r_1}]_{{\scriptscriptstyle \mathbb {Z}}}\subset \Omega $
be the first interval from
$\Sigma $
and, for some integer
$i_0$
,
$(-1)^{i_0+r_1}p(\nu _{r_1})>0$
. Note that, for
$r_1\ge 2$
,
$0\le \nu _1<\dots <\nu _{r_1-1}$
are the points of the set
$\Omega \cap \Sigma _4.$
Then
$(-1)^{i_0+r_1+i}p(\nu _{r_1}+i)\ge 0$
,
$i=0,\dots ,l_{r_1}$
.
Proceeding similarly, let
$0\le \nu _1<\dots <\nu _{r_2-1}$
be the points of the set
$\Omega $
that have been constructed, and the second interval
$I=[k,k+l]_{{\scriptscriptstyle \mathbb {Z}}}\in \Sigma $
is such that
$k=\nu _{r_2}$
or
$k=\nu _{r_2}-1$
. If
$(-1)^{i_0+r_2}p(k)>0$
, then we include in
$\Omega $
the points of the interval
$[\nu _{r_2},\nu _{r_2}+l_{r_2}]_{{\scriptscriptstyle \mathbb {Z}}}$
, where
$\nu _{r_2}=k$
and
$l_{r_2}=l-1$
. Otherwise, we include in
$\Omega $
the points of the interval
$[\nu _{r_2},\nu _{r_2}+l_{r_2}]_{{\scriptscriptstyle \mathbb {Z}}}$
, where
$\nu _{r_2}=k+1$
and
$l_{r_2}=l$
. In both cases,
$|[\nu _{r_2},\nu _{r_2}+l_{r_2}]_{{\scriptscriptstyle \mathbb {Z}}}|=|I|-1$
and
$(-1)^{i_0+r_2+i}p(\nu _{r_2}+i)\ge 0$
,
$i=0,\dots ,l_{r_2}$
.
Continuing in the same way, we complete the construction of the set
$\Omega $
. Then, according to (2.4),
$$\begin{align*}0=\sum_{i=1}^{n+1}(-1)^{i}p(\nu_i)=(-1)^{i_0}\sum_{i=1}^{n+1}|p(\nu_i)|. \end{align*}$$
From this,
$p(\nu _i)=0$
for
$i=1,\dots ,n+1$
. Therefore,
$p\equiv 0$
, completing the proof.
The proof of Corollary 1.6 immediately implies the following result.
Corollary 2.6. If the set
$\{\varphi _k\}_{k=1}^n\subset C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
is a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system,
$p\in L_n$
, and
$N(p)=n-1$
, then the polynomial p has no intervals of the second and third types.
To conclude this section, we give an example of a
$T_0$
-system which is not a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system.
Example 2.7. For
$k=1,\dots ,q-1$
, define
$$\begin{align*}\varphi_k(\nu)= \begin{cases} 0,&\nu=0,\dots,k-2,k,\dots,q-1,\\ 1,&\nu=k-1,q, \end{cases} \quad\varphi_q(\nu)= \begin{cases} 0,&\nu=0,\dots,q-2,\\ 1,&\nu=q-1,\\ -1,&\nu=q. \end{cases} \end{align*}$$
Then
$\dim {L_q}=q$
and
$N_0(\varphi _k)=q-1$
,
$k=1,\dots ,q$
. Note that the function
$\varphi _q(\nu )$
has a zero of the second type at the point q and hence
$N(\varphi _q)=q$
.
For an arbitrary polynomial
$p(\nu )=\sum _{k=1}^qa_k\varphi _k(\nu )$
, the following equalities hold:
$$ \begin{align} p(\nu)= \begin{cases} a_{\nu+1},&\nu=0,\dots,q-1,\\ a_1+\dots+a_{q-1}-a_q,&\nu=q. \end{cases} \end{align} $$
If p has q zeros of the first type, then either all coefficients of the polynomial are zero, or
$q-1$
coefficients are zero and
$a_1+\dots +a_{q-1}-a_q=0$
, that is, in any case all coefficients of p are zeros. Therefore, if p is nontrivial,
$N_0(p)\le q-1$
. Thus, the set
$\{\varphi _k\}_{k=1}^q$
is a
$T_0$
-system but not a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system.
Further, define for any
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
$$\begin{align*}p^{*}(\nu)=\sum_{k=1}^{q-1}(f(k-1)+\delta)\varphi_k(\nu)+(f(q-1)-\delta)\varphi_q(\nu), \end{align*}$$
where
$$\begin{align*}\delta=-\frac{1}{q+1}\Big(\,\sum_{k=1}^{q-1}f(k-1)-f(q-1)-f(q)\Big). \end{align*}$$
By (2.5), we have
$$\begin{align*}p^{*}(\nu)= \begin{cases} f(\nu)+\delta,&\nu=0,\dots,q-2,\\ f(\nu)-\delta,&\nu=q-1,q. \end{cases} \end{align*}$$
If
$\delta =0$
, then
$f\equiv p^{*}\in L_q$
and
$E_q(f,L_q)_{\infty }=0$
. If
$\delta \neq 0$
,
$e(\nu )=p^{*}(\nu )-f(\nu )$
,
$$\begin{align*}\varepsilon_{\nu}= \begin{cases} \operatorname{sign} \delta,&\nu=0,\dots,q-2,\\ -\operatorname{sign} \delta,&\nu=q-1,q, \end{cases} \end{align*}$$
then, by Theorem 1.3, for
$\nu \in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
and all polynomials
$p\in L_q$
$$\begin{align*}\varepsilon_{\nu} e(\nu)=\|e\|_{\infty}=|\delta|,\qquad \sum_{\nu=0}^{q}\varepsilon_{\nu} p(\nu)=0. \end{align*}$$
Consequently,
$p^{*}\in L_q$
is the unique best uniform approximant of the function f and
$E_q(f,L_q)_{\infty }=|\delta |$
. Note that the metric projection operator
$Pf=p^{*}$
onto the q-dimensional subspace
$L_q$
is linear (see, e.g., [Reference Borodin6]), and for any function
$f\not \in L_q$
, the polynomial
$Pf$
does not admit a Chebyshev alternance set of length
$q+1$
.
3 Proofs of Theorems 1.8, 1.10, and 1.11
Proof of Theorem 1.8
Let
$k_{l}>0$
be the leading coefficient of the polynomial
$P_{l}(\lambda )$
. Then, by (1.7), we have
$\beta _{l}k_{l+1}=\rho _{l}k_{l}$
. Using this, for the monic polynomials
$Q_{l}(\lambda )=k_{l}^{-1}P_{l}(\lambda )$
, we obtain the recurrence relation
$$\begin{align*}B_{l}Q_{l-1}(\lambda)+A_{l}Q_{l}(\lambda)+Q_{l+1}(\lambda)=\lambda Q_{l}(\lambda), \end{align*}$$
where
$B_{0}>0$
is arbitrary and
$$ \begin{align} A_{l}=\frac{\alpha_{l}}{\rho_{l}},\quad B_{l}=\frac{\gamma_{l-1}\beta_{l-1}}{\rho_{l-1}\rho_{l}}= \frac{\gamma_{l-1}\rho_{l-1}k_{l-1}^{2}}{\beta_{l-1}\rho_{l}k_{l}^{2}}. \end{align} $$
We have
$A_{l}\in \mathbb {R}$
and
$B_{l}>0$
for any
$l\ge 0$
. These conditions are necessary and sufficient in Favard’s theorem [Reference Chihara8, Ch. II, Th. 6.4] on the existence of a nondecreasing function of bounded variation with infinite spectrum which generates a positive measure
$\mu $
such that
$$\begin{align*}\int_{\mathbb{R}}Q_{l}(\lambda)Q_{m}(\lambda)\,d\mu(\lambda)=B_{0}B_{1}\cdots B_{l}\delta_{lm}. \end{align*}$$
Now, to show (1.8), it is enough to set
$B_{0}=\rho _{0}^{-1}$
.
According to [Reference Chihara8, Ch. IV, Th. 2.2], the measure
$\mu $
has a finite support
$[a,b]$
if and only if the sequences
$A_{l}$
and
$B_{l}$
are uniformly bounded. In this case, the endpoints of the interval
$[a,b]$
will be the limit points of the zeros of
$Q_{l}(\lambda )$
, and, therefore, of
$P_{l}(\lambda )$
as well. By (3.1), the boundedness of
$A_{l}$
and
$B_{l}$
is equivalent to the boundedness of the sequences (1.9), which completes the proof.
Proof of Theorem 1.10
Recall that
$\lambda _{l,l}<\dots <\lambda _{1,l}$
are the zeros of the polynomials
$P_l(\lambda )$
,
$l\in \mathbb {N}$
, and
$\lambda _{q+1}<\dots <\lambda _{1}$
are zeros of the polynomial
$\widetilde {P}_{q+1}(\lambda )$
, cf. (1.10).
In what follows, we will use the fact that the zeros of orthogonal polynomials
$P_{l-1}(\lambda )$
and
$P_l(\lambda )$
,
$l\in [2,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, as well as
$P_q(\lambda )$
and
$\widetilde {P}_{q+1}(\lambda )$
, interlace (see [Reference Szegö35, Ch. III, § 3.3]):
$$ \begin{align} \begin{gathered} \lambda_{l,l}<\lambda_{l-1,l-1}<\lambda_{l-1,l}<\lambda_{l-2,l-1}<\dots< \lambda_{2,l-1}<\lambda_{2,l}<\lambda_{1,l-1}<\lambda_{1,l},\\ \lambda_{q+1}<\lambda_{q,q}<\lambda_{q}<\lambda_{q-1,q}<\dots< \lambda_{2,q}<\lambda_{2}<\lambda_{1,q}<\lambda_{1}. \end{gathered} \end{align} $$
Let the interval
$(a,b)$
contain the zeros of the polynomial
$P_q(\lambda )$
and let
$k\in [1,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. If
$s=0$
, then
$\psi _{k}(0)=P_0(\lambda _{k})=1$
, so further
$s\in [1,q]_{{\scriptscriptstyle \mathbb {Z}}}$
.
We will also use the following observation. If
$s\in [1,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$l\in [1,s]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\lambda _{s+1,s}=a$
,
$\lambda _ {0,s}=b$
, and
$\lambda _{l,s}<\lambda _{k}<\lambda _{l-1,s}$
, then
$$ \begin{align} \operatorname{sign} \psi_k(s)=\operatorname{sign} P_s(\lambda_{k})=(-1)^{l-1}, \end{align} $$
and if
$\lambda _{k}=\lambda _{l,s}$
or
$\lambda _{k}=\lambda _{l-1,s}$
, then
$$ \begin{align} \psi_k(s)=P_s(\lambda_{k})=P_s(\lambda_{l,s})=P_s(\lambda_{l-1,s})=0. \end{align} $$
Further, for
$k=1$
and
$s\in [1,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, the inequality
$\lambda _{1,s}<\lambda _{1}$
yields
$$ \begin{align} \psi_1(s)>0,\quad s\in [0,q]_{{\scriptscriptstyle\mathbb{Z}}}. \end{align} $$
Therefore,
$N(\psi _1)=0$
. For
$k=q+1$
and
$s\in [1,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, from the inequalities (3.2) we have
$\lambda _{q+1}<\lambda _{s,s}$
, so
$\operatorname {sign} \psi _{q+1}(s)=(-1)^{s}$
and
$\psi _{q+1}(0)=1$
. Thus,
$N(\psi _{q+1})=q$
.
Let
$k\in [2,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. Since, according to (3.2),
$\lambda _{k}<\lambda _{k-1,q}$
, there exists a smallest
$i_1\le q$
for which
$$ \begin{align} \lambda_{k,i_1}<\lambda_{k-1,i_1-1}\le\lambda_{k}<\lambda_{k-1,i_1},\quad i_1\ge k. \end{align} $$
If
$i_1\le s\le q$
, then, by (3.2) and (3.6),
$$\begin{align*}\lambda_{k,i_1}\le\lambda_{k,s}\le \lambda_{k,q}< \lambda_{k}<\lambda_{k-1,i_1}\le\lambda_{k-1,s}\le\lambda_{k-1,q}. \end{align*}$$
Using (3.3), we then derive
$$ \begin{align} \operatorname{sign} \psi_k(s)=(-1)^{k-1}. \end{align} $$
Since, according to (3.2),
$\lambda _{k}<\lambda _{k-1,i_1}<\lambda _{k-2,i_1-1}$
, there exists a smallest
$i_2\le i_1-1 $
for which
$$ \begin{align} \lambda_{k-1,i_2}<\lambda_{k-2,i_2-1}\le\lambda_{k}<\lambda_{k-2,i_2}, \quad i_2\ge k-1. \end{align} $$
If
$i_2\le s<i_1-1$
, then, by (3.2), (3.6) and (3.8),
$$\begin{align*}\lambda_{k-1,i_2}\le\lambda_{k-1,s}<\lambda_{k-1,i_1-1}\le\lambda_{k}<\lambda_{k-2,i_2}\le\lambda_{k-2,s}<\lambda_{k-2,i_1-1}. \end{align*}$$
Therefore, using (3.3) and (3.4), we obtain
$$\begin{align*}\operatorname{sign} \psi_k(s)=(-1)^{k-2},\quad \operatorname{sign} \psi_k(i_1-1)=(-1)^{k-2}\vee 0. \end{align*}$$
Proceeding similarly for
$j\in [2,k-1]_{{\scriptscriptstyle \mathbb {Z}}}$
, we construct points
$\{i_j\}_{j=1}^{k-1}$
such that
$$\begin{align*}k-j+1\le i_j\le i_{j-1}-1, \quad \lambda_{k-j+1,i_j}<\lambda_{k-j,i_j-1}\le\lambda_{k}<\lambda_{k-j,i_j}, \end{align*}$$
and for
$i_j\le s<i_{j-1}-1$
$$\begin{align*}\lambda_{k+1-j,i_j}\le\lambda_{k+1-j,s}<\lambda_{k+1-j,i_{j-1}-1}\le\lambda_{k}<\lambda_{k-j,i_j}\le\lambda_{k-j,s}<\lambda_{k-j,i_{j-1}-1}, \end{align*}$$
$$ \begin{align} \operatorname{sign} \psi_k(s)=(-1)^{k-j},\quad \operatorname{sign} \psi_k(i_{j-1}-1)=(-1)^{k-j}\vee 0. \end{align} $$
In the case
$1\le s<i_{k-1}-1$
, we have
$\lambda _{1,s}<\lambda _{1,i_{k-1}-1}\le \lambda _{k}$
. Hence, in light of (3.2),
$$ \begin{align} \operatorname{sign} \psi_k(s)=1,\quad \operatorname{sign} \psi_k(i_{k-1}-1)=1\vee 0. \end{align} $$
If for all
$j\in [2, k-1]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\lambda _{k-j,i_j-1}\neq \lambda _{k}$
, then, according to (3.7), (3.9), (3.10), only the points
$\nu _j=i_j$
,
$j\in [1,k-1]_{{\scriptscriptstyle \mathbb {Z}}}$
, are zeros of
$\psi _k(\nu )$
and, moreover,
$N(\psi _k)=k-1$
.
Let
$\psi _k(i_j-1)=0$
. Since
$\psi _k(i_{j+1})\neq 0$
, it follows that
$i_{j+1}\le i_j-2<i_j-1$
and
$$\begin{align*}\operatorname{sign}{}(\psi_k(i_{j}-2)\psi_k(i_{j}))=(-1)^{k-j-1}(-1)^{k-j}=-1. \end{align*}$$
Therefore, the zero
$\nu _j=i_j$
is replaced by
$\nu _j=i_j-1$
and we again have
$N(\psi _k)=k-1$
. Along the way, we proved that if
$\psi _{k}(\nu )=0$
,
$k\in [2,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, then
$\nu \in [1,q-1]_ {{\scriptscriptstyle \mathbb {Z}}}$
and
$\psi _{k}(\nu -1)\psi _{k}(\nu +1)<0$
. This implies that
$S^{-}(\psi _k)=S^{+}(\psi _k)=k-1$
, completing the proof.
3.1 Proof of Theorem 1.11
Step 1. First, let us rewrite the discrete Sturm-Liouville problems (1.7) and (1.12) and then derive the Christoffel-Darboux formulas.
The self-adjoint form of the problem (1.7) is given by
$$ \begin{align*} \nabla(w_l\Delta P_l(\lambda))+c_lP_l(\lambda)=\lambda d_lP_l(\lambda), \quad l\in \mathbb{Z}_{+},\quad \lambda\in\mathbb{C}, \end{align*} $$
where
$\Delta b_{l}=b_{l+1}-b_{l}$
,
$\nabla b_{l}=b_{l}-b_{l-1}$
are the forward and backward differences, respectively,
$d_l$
is defined in (1.8), and
$$ \begin{align} w_l=\frac{\beta_ld_l}{\rho_l},\quad c_l=\frac{(\alpha_l+\beta_l+\gamma_{l-1})d_l}{\rho_l}. \end{align} $$
Indeed, using (1.7), (1.8) and (3.11), we obtain
$$\begin{align*}w_{l-1}=\frac{\gamma_{l-1}d_l}{\rho_l},\quad \nabla(w_l\Delta P_l(\lambda))=w_lP_{l+1}(\lambda)+w_{l-1}P_{l-1}(\lambda)-(w_{l}+w_{l-1})P_{l}(\lambda), \end{align*}$$
and
$$\begin{align*}\nabla(w_l\Delta P_l(\lambda))+c_lP_l(\lambda)-\lambda d_lP_l(\lambda) =\frac{d_l}{\rho_l}\,\bigl\{\beta_lP_{l+1}(\lambda)+\alpha_lP_{l}(\lambda)+\gamma_{l-1}P_{l-1}(\lambda)-\lambda\rho_lP_{l}(\lambda)\bigr\}. \end{align*}$$
Now we can easily derive the Christoffel-Darboux formula
$$ \begin{align} \sum_{s=0}^{l}d_{s}P_{s}(x)P_{s}(y)=w_{l}\,\frac{P_{l+1}(x)P_{l}(y)-P_{l}(x)P_{l+1}(y)}{x-y},\quad x\neq y. \end{align} $$
To see this, using the Lagrange formula [Reference Agarwal, Bohner, Grace and O’Regan2]
$$\begin{align*}(x-y)d_sP_s(x)P_s(y)=P_s(y)\nabla(w_s\Delta P_s(x))-P_s(x)\nabla(w_s\Delta P_s(y)), \end{align*}$$
we have
$$ \begin{align*}& (x-y)\sum_{s=0}^{l}d_sP_s(x)P_s(y)=\sum_{s=0}^{l}\bigl\{P_s(y)\nabla(w_s\Delta P_s(x))-P_s(x)\nabla(w_s\Delta P_s(y))\bigr\}\\ &=\sum_{s=0}^{l}\bigl\{w_sP_{s+1}(x)P_s(y)+w_{s-1}P_s(y)P_{s-1}(x)-w_sP_{s+1}(y)P_s(x)-w_{s-1}P_{s}(x)P_{s-1}(y)\bigr\}\\ &=w_l\bigl\{P_{l+1}(x)P_l(y)-P_{l}(x)P_{l+1}(y)\bigr\}. \end{align*} $$
Further, by (3.12), it follows that
$w_{l}=k_{l}d_{l}/k_{l+1}$
, where
$k_{l}$
is the leading coefficient of the polynomial
$P_{l }$
. Applying (1.8) and (3.11), we get
$$\begin{align*}k_{0}=1,\quad k_{l}=\frac{\rho_0\cdots\rho_{l-1}}{\beta_0\cdots\beta_{l-1}}, \quad l\ge 1. \end{align*}$$
Moreover, the Sturm-Liouville problem (1.12) can also be written in an equivalent self-adjoint form as follows:
$$ \begin{align} \begin{gathered} \nabla(w_l\Delta \psi(\nu))+c_l\psi(\nu)=\lambda d_l\psi(\nu),\\ \psi(-1)=0,\quad\psi(q+1)-\eta\psi(q)=0. \end{gathered} \end{align} $$
From (3.13), as in the proof of (3.12), we obtain the following Lagrange equality:
$$ \begin{align*} d_{\nu}(\lambda_l-\lambda_k)\psi_l(\nu)\psi_k(\nu)= \nabla(w_{\nu}\{\psi_l(\nu)\Delta \psi_k(\nu)-\psi_k(\nu)\Delta \psi_l(\nu)\}). \end{align*} $$
It is important to mention that the Christoffel-Darboux formula for the eigenfunctions
$\psi _k$
has the form
$$ \begin{align} \sum_{\nu=0}^{l}d_{\nu}\psi_k(\nu)\psi_s(\nu)=w_{l}\,\frac{\psi_k(l+1)\psi_s(l)-\psi_k(l)\psi_s(l+1)}{\lambda_k-\lambda_s}, \end{align} $$
which immediately implies the discrete orthogonality of the eigenfunctions. Indeed, for
$k\neq s$
, due to the boundary condition in (1.12),
$$ \begin{align*} (\psi_k,\psi_s)&=\sum_{\nu=0}^{q}d_{\nu}\psi_k(\nu)\psi_s(\nu)=w_{q}\,\frac{\psi_k(q+1)\psi_s(q)-\psi_k(q)\psi_s(q+1)}{\lambda_k-\lambda_s}\\ &=w_{q}\,\frac{(\psi_k(q+1)-\eta\psi_k(q))\psi_s(q)-(\psi_s(q+1)-\eta\psi_s(q))\psi_k(q)}{\lambda_k-\lambda_s}=0. \end{align*} $$
Step 2. Now we will prove several lemmas on zeros and sign changes of discrete functions.
Lemma 3.1. If
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
, then
$$ \begin{align} S^{-}(f)\le N(f)\le S^{+}(f). \end{align} $$
Proof. It suffices to compare
$S^{-}(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})$
,
$N(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})$
, and
$S ^{+}(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})$
at interior points of the intervals
$[m,n]_{{\scriptscriptstyle \mathbb {Z}}}$
, where f equals zero. We may assume that
$n-m\ge 2$
.
-
○ If
$m=0$
,
$f(0)=f(1)=\dots =f(n-1)=0$
, and
$f(n)\neq 0$
, then
$S^{-}(f ,[0,n]_{{\scriptscriptstyle \mathbb {Z}}})=0$
,
$N(f,[0,n]_{{\scriptscriptstyle \mathbb {Z}}})=n$
,
$S^{+}(f,[0,n] _{{\scriptscriptstyle \mathbb {Z}}})=n$
and thus (3.15) is valid. -
○ If
$n=q$
,
$f(m)\neq 0$
,
$f(m+1)=\dots =f(q)=0$
, and
$f(n)\neq 0$
, then
$S^ {-}(f,[m,q]_{{\scriptscriptstyle \mathbb {Z}}})=0$
,
$N(f,[m,q]_{{\scriptscriptstyle \mathbb {Z}}})=q-m$
,
$S^{+}(f, [m,q]_{{\scriptscriptstyle \mathbb {Z}}})=q-m$
and (3.15) follows. -
○ If
$f(m)f(n)<0$
(for example,
$f(m)>0$
,
$f(m+1)=\dots =f(n-1)=0$
,
$f(n )<0$
), then
$S^{-}(f,[m,q]_{{\scriptscriptstyle \mathbb {Z}}})=1$
,
$N(f,[m,q]_{{\scriptscriptstyle \mathbb {Z}}})=n-m-1 $
,
$S^{+}(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})=n-m-1$
for even
$n-m$
and
$S^{+}(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})=n-m$
for odd
$n-m$
. Hence, (3.15) is valid. -
○ If
$f(m)f(n)>0$
(for example,
$f(m)>0$
,
$f(m+1)=\dots =f(n-1)=0$
,
$f(n )>0$
), then
$S^{-}(f,[m,q]_{{\scriptscriptstyle \mathbb {Z}}})=1$
,
$N(f,[m,q]_{{\scriptscriptstyle \mathbb {Z}}})=n-m-1 $
,
$S^{+}(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})=n-m-1$
for odd
$n-m$
and
$S^{+}(f,[m,n]_{{\scriptscriptstyle \mathbb {Z}}})=n-m$
for even
$n-m$
, implying (3.15).
The proof is now complete.
The next few results correspond to discrete analogues of Rolle’s theorem. Below, for integers
$m<n$
, we say
$(m,n)$
is a pair of generalized sign change of f if
$f(m)f(n)<0$
and
$f(m+1)=\ldots =f(n-1)=0$
. Clearly, an ordinary sign-change pair is also generalized (for
$n-m=1$
). Note that the number of pairs of generalized sign changes of f on
$[a,b]_{{\scriptscriptstyle \mathbb {Z}}}$
is exactly
$S^{-}(f,[a,b]_{{\scriptscriptstyle \mathbb {Z}}})$
.
Lemma 3.2. If
$m<n$
and
$f(m)f(n)<0$
, then the function f has a zero on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
and a generalized sign change on
$[m,n]_{{\scriptscriptstyle \mathbb {Z}}}$
.
Proof. We can assume that
$f(m)<0$
and
$f(n)>0$
. If
$n=m+1$
, then n is a zero of the second type and
$(m,n)$
is the pair of a sign change. Let
$n>m+1$
and
$s_1\le n$
be the largest integer such that
$f(m)<0,f(m+1)<0,\dots ,f(s_1-1)<0 $
. Then either
$f(s_1)>0$
or
$f(s_1)=0$
. If
$f(s_1)>0$
, then
$s_1\ge m+1$
is a zero of the second type and
$(s_1-1,s_1)$
is the pair of a sign change. Suppose
$f(s_1)=0$
and
$s_2<n$
is the largest integer such that
$f(s_1)=f(s_1+1)=\dots =f(s_2-1)=0$
. Then either
$f(s_2)>0$
or
$f(s_2)<0$
. If
$f(s_2)>0$
, then
$s_1$
,
$s_1+1$
,…,
$s_2-1$
are zeros of the first type and
$(s_1-1,s_2)$
is the pair of a generalized sign change. If
$f(s_2)<0$
, using similar arguments, we arrive at the required statement.
Lemma 3.3. Let
$m<n$
be zeros of f. Then the differences
$\nabla f$
on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$\Delta f$
on
$[m,n-1 ]_{{\scriptscriptstyle \mathbb {Z}}}$
have zeros.
Proof. First, let us prove the lemma for
$\nabla f$
. Consider the possible values f at the zeros m and n. We will repeatedly use the trivial formula
$$ \begin{align} f(l)-f(k)=\sum_{\nu=k+1}^{l}\nabla f(\nu). \end{align} $$
-
○ If
$f(m)=f(n)=0$
, then according to (3.16), either
$\nabla f (\nu )=0$
for all
$\nu \in [m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
, or
$\nabla f(\nu _1)\nabla f(\nu _2)<0 $
for some
$\nu _1,\nu _2\in [m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
, and by Lemma 3.2
$\nabla f$
has a zero on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
. -
○ If
$f(m)=0$
,
$f(n-1)>0$
,
$f(n)<0$
, then
$f(n-1)-f(m)>0$
,
$\nabla f(n)<0$
. According to (3.16), for some
$s\in [m+1,n-1]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\nabla f(s)>0$
, so
$\nabla f(s)\nabla f(n)<0$
. By Lemma 3.2,
$\nabla f$
on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
has a zero. The case
$f(m)=0$
,
$f(n-1)<0$
,
$f(n)>0$
is considered similarly. -
○ If
$f(m-1)>0$
,
$f(m)<0$
,
$f(n)=0$
, then
$\nabla f(m)<0$
,
$f(n)-f( m)>0$
. According to (3.16), for some
$s\in [m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\nabla f(s)>0$
, therefore
$\nabla f(m)\nabla f (s)<0$
. In light of Lemma 3.2,
$\nabla f$
on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
has a zero. The case
$f(m-1)<0$
,
$f(m)>0$
,
$f(n)=0$
is treated similarly. -
○ If
$f(m-1)>0$
,
$f(m)<0$
,
$f(n-1)>0$
,
$f(n)<0$
, then
$\nabla f(m)< 0$
,
$f(n-1)-f(m)>0$
. According to (3.16), for some
$s\in [m+1,n-1]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\nabla f(s)>0$
, so
$\nabla f(m)\nabla f(s)<0$
. By Lemma 3.2,
$\nabla f$
on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
has a zero. -
○ If
$f(m-1)>0$
,
$f(m)<0$
,
$f(n-1)<0$
,
$f(n)>0$
, then
$\nabla f(m)< 0$
,
$\nabla f(n)<0$
. Using again Lemma 3.2,
$\nabla f$
on
$[m+1,n]_{{\scriptscriptstyle \mathbb {Z}}}$
has a zero. The remaining two cases are considered similarly.
Finally, using the equality
$ f(l)-f(k)=\sum _{\nu =k}^{l-1}\Delta f(\nu ) $
one similarly proves the statement for
$\Delta f$
. The proof is now complete.
Lemma 3.4. Let
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$0\le m<n\le m'<n'\le q$
, and the pairs
$(m,n)$
and
$(m',n')$
be consecutive generalized sign changes of f. Then the differences
$\nabla f$
on
$[m+1,m'+1]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$\Delta f$
on
$[n-1,n'-1]_{{\scriptscriptstyle \mathbb {Z}}}$
have generalized sign changes.
Proof. Let
$f(m)<0$
and
$f(n)>0$
. Then
$$\begin{align*}f(\nu)=0,\quad \nu\in [m+1,n-1]_{{\scriptscriptstyle\mathbb{Z}}},\quad f(\nu)>0,\quad \nu\in [n,m']_{{\scriptscriptstyle\mathbb{Z}}}, \end{align*}$$
$$\begin{align*}f(\nu)=0,\quad \nu\in [m'+1,n'-1]_{{\scriptscriptstyle\mathbb{Z}}},\quad f(n')<0. \end{align*}$$
Since
$\nabla f(m+1)=f(m+1)-f(m)>0$
and
$\nabla f(m'+1)=f(m'+1)-f(m')<0 $
, by Lemma 3.2,
$\nabla f$
has a generalized sign change on
$[m+1,m'+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. The case
$f(m)>0$
and
$f(n)<0$
and the fact that the difference
$\Delta f$
has a generalized sign change on
$[n-1,n'-1]_{{\scriptscriptstyle \mathbb {Z}}}$
can be obtained similarly.
Lemma 3.5. Let
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$f(-1)=f(q+1)=0$
,
$0\le m<n\le q$
, and the pair
$(m,n)$
be a generalized sign change of f. Then the difference
$\nabla f$
on
$[0,m+1]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$[n,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
has generalized sign changes.
Proof. Let
$f(\nu )=0$
for
$\nu \in [-1,m-1]_{{\scriptscriptstyle \mathbb {Z}}}$
. If
$f(m)>0$
and
$f(n)<0$
, then
$\nabla f(m)>0$
. Since
$\nabla f(m+1)<0$
,
$(m,m+1)$
is the pair of a sign change of
$\nabla f$
. The case
$f(m)<0$
and
$f(n)>0$
is treated similarly.
Suppose the function
$f(\nu )$
has a nonzero value on
$[0,m-1]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$m\ge 1$
, for example,
$f(0)>0$
. If
$f(m)>0$
and
$f(n)<0$
, then
$\nabla f(0)>0$
and
$\nabla f(m+1)<0$
. By Lemma 3.2,
$\nabla f$
has a generalized sign change on
$[0,m+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. If
$f(m)<0$
and
$f(n)>0$
, then
$f(m)-f(0)<0$
. Then for some
$\nu _1\in [1,m]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$\nabla f(\nu _1)<0$
and
$\nabla f(0)>0$
. By Lemma 3.2,
$\nabla f$
has a generalized sign change on
$[0,m]_{{\scriptscriptstyle \mathbb {Z}}}$
. The case
$f(0)<0$
is treated similarly. Therefore,
$\nabla f$
on
$[0,m+1]_{{\scriptscriptstyle \mathbb {Z}}}$
always has a generalized sign change.
The fact that the difference
$\nabla f$
has a generalized sign change on
$[n,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
can be shown similarly. The proof is now complete.
Definition 3.6. By
$K({\mskip 2mu\cdot \mskip 2mu})$
we denote any one of the following quantities:
$N({\mskip 2mu\cdot \mskip 2mu})$
,
$S^{-}({\mskip 2mu\cdot \mskip 2mu})$
, or
$S^{+}({\mskip 2mu\cdot \mskip 2mu})$
.
We will use the simple facts that
$K(f,[a,b]_{{\scriptscriptstyle \mathbb {Z}}})=K(f,[a,c]_{{\scriptscriptstyle \mathbb {Z}}})+K(f,[c,b]_{{\scriptscriptstyle \mathbb {Z}}})$
whenever
$a\le c\le b$
with
$f(c)\ne 0$
, and that
$K(fg)=K(f)$
if g preserves the sign.
Proposition 3.7. If
$f\in C[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$f\ne 0$
, then
$$ \begin{align} K(\nabla f,[1,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge K(f,[0,q]_{{\scriptscriptstyle\mathbb{Z}}})-1, \end{align} $$
$$ \begin{align} K(\Delta f,[0,q-1]_{{\scriptscriptstyle\mathbb{Z}}})\ge K(f,[0,q]_{{\scriptscriptstyle\mathbb{Z}}})-1. \end{align} $$
If, in addition,
$f(-1)=f(q+1)=0$
, then
$$ \begin{align} K(\nabla f,[0,q+1]_{{\scriptscriptstyle\mathbb{Z}}})\ge K(f,[0,q]_{{\scriptscriptstyle\mathbb{Z}}})+1. \end{align} $$
Remark 3.8. Note that (3.18) was proved in [Reference Hartman19, Proposition 5.1] (see also the monograph [Reference Agarwal1, Theorem 1.8.1]) for
$K({\mskip 2mu\cdot \mskip 2mu})=N({\mskip 2mu\cdot \mskip 2mu})$
.
Proof of Proposition 3.7
As usual, we denote
$K(f)= K(f,[0,q]_{{\scriptscriptstyle \mathbb {Z}}}) .$
The cases
$f>0$
and
$f<0$
on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
are straightforward and may be omitted.
(a) The cases
$K({\mskip 2mu\cdot \mskip 2mu})=N({\mskip 2mu\cdot \mskip 2mu})$
or
$K({\mskip 2mu\cdot \mskip 2mu})=S^{-}({\mskip 2mu\cdot \mskip 2mu})$
. To prove inequality (3.17) in the first case, let
$\{\nu _1<\dots <\nu _{s}\}\subset [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
be the zeros of f. By Lemma 3.3 on the intervals
$[\nu _1+1,\nu _2]_{{\scriptscriptstyle \mathbb {Z}}},\dots ,[\nu _{s-1}+1,\nu _{s}]_{{\scriptscriptstyle \mathbb {Z}}}$
, the function
$\nabla f$
has at least
$s-1$
zeros. Therefore,
$N(\nabla f,[1,q]_{{\scriptscriptstyle \mathbb {Z}}})\ge N(f)-1$
.
In the second case, let
$\{(m_i,n_i)\}_{i=1}^{s}\subset [0,q]_{{\scriptscriptstyle \mathbb {Z}}}\times [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$m_i<n_i\le m_{i+1}<n_{i+1}$
, be the generalized sign changes of f on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. By Lemma 3.4,
$\nabla f$
has generalized sign changes on the intervals
$[m_1+1,m_2+1]_{{\scriptscriptstyle \mathbb {Z}}},\dots ,[m_{s-1}+1,m_s+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. Since this number is not less than
$s-1$
,
$S^{-}(\nabla f,[1,q]_{{\scriptscriptstyle \mathbb {Z}}})\ge S^{-}(f)-1$
.
Inequality (3.18) can be established similarly.
Suppose
$f(-1)=f(q+1)=0$
. By Lemma 3.3, the difference
$\nabla f$
has at least two additional zeros on
$[0,\nu _{1}]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$[\nu _{s}+1,q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. Similarly, by Lemma 3.5,
$\nabla f$
has at least two generalized sign changes on the intervals
$[0,m_{1}+1]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$[\nu _{s},q+1]_{{\scriptscriptstyle \mathbb {Z}}}$
. This establishes (3.19).
(b) The case
$K({\mskip 2mu\cdot \mskip 2mu})=S^{+}({\mskip 2mu\cdot \mskip 2mu})$
. For the sake of brevity, we will prove the intermediate inequality
$$ \begin{align} S^{+}(\nabla f)\ge S^{+}(f),\quad f(-1)=0, \end{align} $$
by induction on q. Inequalities (3.17) and (3.19) can be proved in a similar way.
Let
$n=q=1$
and
$f(0)=\nabla f(0)>0$
. If
$f(1)>0$
, then
$S^{+}(\nabla f)=0$
and (3.20) is valid. If
$f(1)\le 0$
, then
$\nabla f(1)<0$
, so
$S^{+}(\nabla f)=S^{+}(f)=1$
and (3.20) holds. The case
$f(0)<0$
can be treated similarly. Let
$f(0)=\nabla f(0)=0$
. If
$f(1)\neq 0$
, then
$S^{+}(\nabla f)=S^{+}(f)=1$
and (3.20) is valid.
Assume that inequality (3.20) holds for all
$n\le q-1$
. To prove it for
$n=q$
, if
$n\le q-1$
is chosen so that
$f(n)\neq 0$
,
$\nabla f(n)\neq 0$
and
$S^{+}(\nabla f,[n,q ]_{{\scriptscriptstyle \mathbb {Z}}})\ge S^{+}(f,[n,q]_{{\scriptscriptstyle \mathbb {Z}}})$
, then by the inductive assumption we obtain (3.20) as follows:
$$\begin{align*}S^{+}(\nabla f)=S^{+}(\nabla f,[0,n]_{{\scriptscriptstyle\mathbb{Z}}})+S^{+}(\nabla f,[n,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge S^{+}(f,[0,n]_{{\scriptscriptstyle\mathbb{Z}}})+S^{+}(f,[n,q]_{{\scriptscriptstyle\mathbb{Z}}})=S^{+}(f). \end{align*}$$
-
○ Let
$f(q-1)>0$
. If
$f(q)>0$
, then, by the inductive assumption,
$$\begin{align*}S^{+}(\nabla f)\ge S^{+}(\nabla f,[0,q-1]_{{\scriptscriptstyle\mathbb{Z}}})\ge S^{+}(f,[0,q-1]_{{\scriptscriptstyle\mathbb{Z}}})=S^{+}(f). \end{align*}$$
-
○ Let
$f(q-1)>0$
,
$f(q) \le 0$
, and
$s \le q+1$
be such that
$$\begin{align*}f(q-s)\le 0,\ f(q-s+1)>0,\ \dots,\ f(q-1)>0. \end{align*}$$
Since
$\nabla f(q-s+1)>0$
and
$\nabla f(q)<0$
, Lemma 3.2 implies that
$S^{+}(\nabla f,[q-s+1,q]_{{\scriptscriptstyle \mathbb {Z}}}) \ge 1$
and
$$\begin{align*}S^{+}(\nabla f,[q-s+1,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge 1=S^{+}(f,[q-s+1,q]_{{\scriptscriptstyle\mathbb{Z}}}). \end{align*}$$
The case
$f(q-1)<0$
can be treated similarly.
Assume now that
$f(q-1)=\dots =f(q-s_1+1)=0$
and
$$\begin{align*}\ f(q-s_1)>0,\ \dots,\ f(q-s_1-s_2+1)>0,\ f(q-s_1-s_2)\le 0. \end{align*}$$
-
○ If
$f(q)=0$
, then
$$\begin{align*}S^{+}(\nabla f,[q-s_1-s_2+1,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge S^{+}(f,[q-s_1-s_2+1,q]_{{\scriptscriptstyle\mathbb{Z}}})=s_1. \end{align*}$$
-
○ If
$f(q)>0$
, then
$$\begin{align*}S^{+}(\nabla f,[q-s_1-s_2+1,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge S^{+}(f,[q-s_1-s_2+1,q]_{{\scriptscriptstyle\mathbb{Z}}})= \begin{cases} s_1-1,& s_1\text{ odd},\\ s_1,& s_1\text{ even}. \end{cases} \end{align*}$$
-
○ If
$f(q)<0$
, then
$$\begin{align*}S^{+}(\nabla f,[q-s_1-s_2+1,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge S^{+}(f,[q-s_1-s_2+1,q]_{{\scriptscriptstyle\mathbb{Z}}})= \begin{cases} s_1,&s_1\text{ odd},\\ s_1-1,&s_1\text{ even}. \end{cases} \end{align*}$$
The remaining cases can be reduced to the ones above by replacing f with
$-f$
. The proof is now complete.
In order to prove Theorem 1.11, we employ the discrete version of the Liouville method (see, e.g., [Reference Bérard and Helffer5] for the continuous case). In addition to the polynomial
$ V(\nu )=\sum _{k=m}^{n}a_k\psi _k(\nu )$
(see (1.13)), we define
$$ \begin{align*} V_r(\nu)=\sum_{s=m}^{n}(\lambda_1-\lambda_s)^ra_s\psi_s(\nu), \quad r\in\mathbb{Z}. \end{align*} $$
Lemma 3.9. We have
$$\begin{align*}K(V_1)\ge K(V). \end{align*}$$
Proof. Multiplying both sides of (3.14) with
$k=1$
by
$(\lambda _1-\lambda _s)a_s$
and summing over s, for
$l\in [ 0,q-1]_{{\scriptscriptstyle \mathbb {Z}}}$
, we derive
$$ \begin{align} g(l)&=\sum_{\nu=0}^ld_{\nu}\psi_1(\nu)V_{1}(\nu) =w_{l}\{\psi_1(l+1)V(l)-\psi_1(l)V(l+1)\}\notag\\ &=-w_{l}\psi_1(l)\psi_1(l+1)\Big\{\frac{V(l+1)}{\psi_1(l+1)}-\frac{V(l)}{\psi_1(l)}\Big\}= -w_{l}\psi_1(l)\psi_1(l+1)\Delta\Big(\frac{V(l)}{\psi_1(l)}\Big). \end{align} $$
Due to the boundary condition in (1.12), we have
$g(-1)=0$
and
$$ \begin{align*} g(q)&=w_{q}\{\psi_1(q+1)V(q)-\psi_1(q)V(q+1)\}\\&=w_{q}\{(\psi_1(q+1)-\eta \psi_1(q))V(q)-\psi_1(q)(V(q+1)-\eta V(q))\}=0. \end{align*} $$
Applying Proposition 3.7, formula (3.21), and taking into account the positivity of
$\psi _1(l)$
(see (3.5)),
$d_l$
,
$w_l$
, we obtain
$$ \begin{align*} K(V_1)&=K\Big(\frac{\nabla g(l)}{d_l\psi_1(l)}\Big)=K(\nabla g,[0,q]_{{\scriptscriptstyle\mathbb{Z}}})\ge K(g,[0,q-1]_{{\scriptscriptstyle\mathbb{Z}}})+1\\ &=K\Big(-w_s\psi_1(l)\psi_1(l+1)\Delta\Big(\frac{V(l)}{\psi_1(l)}\Big),[0,q-1]_{{\scriptscriptstyle\mathbb{Z}}}\Big)+1\\ &=K\Big(\Delta\Big(\frac{V(l)}{\psi_1(l)}\Big),[0,q-1]_{{\scriptscriptstyle\mathbb{Z}}}\Big)+1\ge K\Big(\frac{V(l)}{\psi_1(l)},[0,q]_{{\scriptscriptstyle\mathbb{Z}}}\Big)=K(V), \end{align*} $$
completing the proof of the lemma.
Step 3 of the proof of Theorem 1.11
Note that in the case
$m=n=1$
Theorem 1.11 holds since
$\psi _1$
is positive. Thus, we may assume that
$n\ge 2$
. Moreover, the lower bound in (1.14) for
$m=1$
holds trivially.
Applying Lemma 3.9 repeatedly, we obtain
$$\begin{align*}K(V_r/(\lambda_{1}-\lambda_n)^r)=K(V_r)\ge K(V),\quad r\ge 1,\quad m\ge 1, \end{align*}$$
$$\begin{align*}K(V_r/(\lambda_{1}-\lambda_m)^r)=K(V_r)\le K(V),\quad r\le-1,\quad m\ge 2. \end{align*}$$
Since
$$\begin{align*}\lim_{r\to +\infty}\|\psi_{n}-V_r/(\lambda_{1}-\lambda_n)^r)\|_{\infty}= \lim_{r\to -\infty}\|\psi_{m}-V_r/(\lambda_{1}-\lambda_m)^r)\|_{\infty}=0, \end{align*}$$
by Theorem 1.10 for sufficiently large
$N>0$
we deduce
$$\begin{align*}K(\psi_{n})&=K(V_r/(\lambda_{1}-\lambda_n)^r),\quad r>N,\\K(\psi_{m})&=K(V_r/(\lambda_{1}-\lambda_m)^r),\quad r<-N. \end{align*}$$
Thus,
$$\begin{align*}n-1=K(\psi_{n})=\lim_{r\to+\infty}K(V_r/(\lambda_{1}-\lambda_n)^r)=\lim_{r\to+\infty}K(V_r)\ge K(V), \end{align*}$$
$$\begin{align*}m-1=K(\psi_{m})=\lim_{r\to -\infty}K(V_r/(\lambda_{1}-\lambda_m)^r)=\lim_{r\to-\infty}K(V_r)\le K(V). \end{align*}$$
This and Lemma 3.1 complete the proof.
4 Proof of Theorem 1.14
Let
$\nu \in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$1\le \nu _1<\dots <\nu _{m}\le q$
, and
$$\begin{align*}R_{m}(\nu)=R_{m}(\nu,\nu_1,\dots,\nu_{m})=\prod_{j=1}^{m}(\nu_j-\nu). \end{align*}$$
Let also
$\{\varphi _k\}_{k=1} ^{m+1}$
be a
$T_{{\scriptscriptstyle \mathbb {Z}}}$
-system on
$[0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
and
$$ \begin{align} D_{m+1}(\nu)=D_{m+1}(\nu,\nu_1,\dots,\nu_{m}) =\Delta\left( \begin{matrix} \varphi_1, & \dots, &\varphi_{m},& \varphi_{m+1}\\ \nu_{m}, & \dots, &\nu_1, &\nu \end{matrix}\right). \end{align} $$
Observe that
$\nu _j$
,
$j=1,\dots ,m$
, are the only zeros of the polynomial
$D_{m+1}$
. By applying Theorem 1.4 and basic properties of determinants, we obtain the precise distribution of the signs of
$D_{m+1}(\nu )$
.
Lemma 4.1. For all
$\nu \in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$$ \begin{align*} \operatorname{sign} D_{m+1}(\nu)=\operatorname{sign} D_{m+1}(0,q-m+1,\dots,q)\operatorname{sign} R_{m}(\nu). \end{align*} $$
Proof of Theorem 1.14
Let the zeros of
$P_{q}(\lambda )$
lie on the interval
$(a,b)$
and let
$\eta _{b}=P_{q+1}(b)/P_{q}(b)$
. Then the zeros of the polynomial
$\widetilde {P}_{q+1}(\lambda )=P_{q+1}(\lambda )-\eta P_{q}(\lambda )$
are such that
$\lambda _2,\dots ,\lambda _q\in (a,b)$
,
$\lambda _1<b$
for
$\eta <\eta _b$
,
$\lambda _1=b$
for
$\eta =\eta _b$
, and
$\lambda _1>b$
for
$\eta>\eta _b$
(see [Reference Szegö35, Ch. III, § 3.3]).
(b) In this case
$m=0$
. From the interlacing of zeros of
$P_{\nu }(\lambda )$
(see (3.2)), there holds
$$\begin{align*}\lambda_{1,1}<\dots<\lambda_{1,q}<\lambda_{1}, \end{align*}$$
and therefore,
$P_{\nu }(\lambda _1)>0$
,
$\nu \in [0,q]_{{\scriptscriptstyle \mathbb {Z}}}$
. Applying the Christoffel-Darboux formula (3.12), we obtain
$$ \begin{align*} \sum_{l=0}^{q}d_{l}P_{l}(\lambda_1)P_l(\lambda)&=w_q\,\frac{P_{q+1}(\lambda)P_q(\lambda_1)-P_{q+1}(\lambda_1)P_q(\lambda)}{\lambda-\lambda_1}\\& =w_q\,\frac{\widetilde{P}_{q+1}(\lambda)P_q(\lambda_1)-\widetilde{P}_{q+1}(\lambda_1)P_{q}(\lambda)}{\lambda-\lambda_1} =w_qP_q(\lambda_1)\,\frac{\widetilde{P}_{q+1}(\lambda)}{\lambda-\lambda_1}. \end{align*} $$
Therefore, the coefficients
$a_{\nu }$
in expansion (1.15) satisfy
$$ \begin{align} a_{\nu}=\frac{P_{\nu}(\lambda_1)}{w_qP_q(\lambda_1)}>0,\quad \nu\in [0,q]_{{\scriptscriptstyle\mathbb{Z}}}. \end{align} $$
For
$\eta =\eta _b$
, we have
$\lambda _1=b$
and (1.17) is valid.
(a, c): the case
$m=0$
and
$\eta \neq \eta _b$
. By the Christoffel-Darboux formula,
$$ \begin{align*} \frac{P_{\nu}(\lambda_1)}{P_{\nu}(b)}-\frac{P_{\nu+1}(\lambda_1)}{P_{\nu+1}(b)}= \frac{b-\lambda_1}{w_{\nu}P_{\nu}(b)P_{\nu+1}(b)}\sum_{l=0}^{\nu}d_{l}P_{l}(b)P_{l}(\lambda_1). \end{align*} $$
Since
$\operatorname {sign}{}(b-\lambda _1)=\operatorname {sign}{}(\eta _b-\eta )$
, we have, for
$\nu \in [0,q-1]_{{\scriptscriptstyle \mathbb {Z}}}$
,
$$\begin{align*}\operatorname{sign} \Big(\frac{P_{\nu}(\lambda_1)}{P_{\nu}(b)}-\frac{P_{\nu+1}(\lambda_1)}{P_{\nu+1}(b)}\Big)=\operatorname{sign}{}(\eta_b-\eta). \end{align*}$$
From this and (4.2) we obtain (1.16) and (1.18) for
$m=0$
.
(a): the case
$m\ge 1$
and
$\eta \neq \eta _b$
. Let us calculate the coefficients
$a_{\nu }$
in expansion (1.15) in terms of
$D_{m+1}(\nu ,\nu _1,\dots ,\nu _m)$
,
$1 \le \nu _1<\dots <\nu _{m}\le q$
.
Letting
$\omega _{m+1}(\lambda )=\prod _{j=1}^{m+1}(\lambda -\lambda _j)$
, we have
$$\begin{align*}\frac{\widetilde{P}_{q+1}(\lambda)}{\omega_{m+1}(\lambda)} =\sum_{i=1}^{m+1}\frac{\widetilde{P}_{q+1}(\lambda)}{\omega_{m+1}'(\lambda_i)(\lambda-\lambda_i)}, \end{align*}$$
where
$$ \begin{align*}\omega_{m+1}'(\lambda_i)=\prod_{j\neq i}(\lambda_i-\lambda_j),\quad \operatorname{sign} \omega_{m+1}'(\lambda_i)=(-1)^{i-1}.\end{align*} $$
Applying the Christoffel-Darboux formula, we obtain as above
$$ \begin{align} \sum_{\nu=0}^qd_{\nu}P_{\nu}(\lambda_i)P_{\nu}(\lambda)&=w_q\,\frac{P_{q+1}(\lambda)P_q(\lambda_i)-P_{q}(\lambda)P_{q+1}(\lambda_i)}{\lambda-\lambda_i}\notag \\ &=w_q\,\frac{\widetilde{P}_{q+1}(\lambda)P_q(\lambda_i)}{\lambda-\lambda_i}. \end{align} $$
Therefore,
$$\begin{align*}\frac{\widetilde{P}_{q+1}(\lambda)}{\omega_{m+1}(\lambda)}=\frac{1}{w_q}\sum_{\nu=0}^{q-m}d_{\nu} \sum_{i=1}^{m+1}\frac{P_{\nu}(\lambda_i)}{\omega_{m+1}'(\lambda_i)P_{q}(\lambda_i)}\,P_{\nu}(\lambda) \end{align*}$$
and
$$ \begin{align} a_{\nu}=\frac{1}{w_q}\sum_{i=1}^{m+1}\frac{P_{\nu}(\lambda_i)}{\omega_{m+1}'(\lambda_i)P_{q}(\lambda_i)}. \end{align} $$
Since
$$\begin{align*}\Delta(\lambda_1,\dots,\lambda_{m+1})=\prod_{1\le j<s\le m+1}(\lambda_s-\lambda_j) \end{align*}$$
is the Vandermonde determinant, we derive
$$ \begin{align*} \frac{1}{\omega_{m+1}'(\lambda_i)}& =\frac{1}{\Delta(\lambda_1,\dots,\lambda_{m+1})}\, \frac{\Delta(\lambda_1,\dots,\lambda_{m+1})}{\prod_{j\neq i}(\lambda_i-\lambda_j)} \\ &=\frac{(-1)^{i-1}\Delta(\lambda_1,\dots,\lambda_{i-1},\lambda_{i+1},\dots,\lambda_{m+1})}{\Delta(\lambda_1,\dots,\lambda_{m+1})}. \end{align*} $$
Taking into account (4.4), this gives
We now rewrite formula (1.7) as follows:
$$\begin{align*}P_{l-1}(\lambda)=(A_l\lambda+B_l)P_l(\lambda)-C_lP_{l+1}(\lambda), \end{align*}$$
where
$$\begin{align*}A_l=\frac{\rho_l}{\gamma_{l-1}}>0,\quad B_l=-\frac{\alpha_l}{\gamma_{l-1}},\quad C_l=\frac{\beta_l}{\gamma_{l-1}}>0. \end{align*}$$
This implies the recurrence relation
$$\begin{align*}\frac{P_{q-j}(\lambda_i)}{P_{q}(\lambda_i)}=(A_{q-j+1}\lambda_i+B_{q-j+1})\,\frac{P_{q-j+1}(\lambda_i)}{P_{q}(\lambda_i)}- C_{q-j+1}\,\frac{P_{q-j+2}(\lambda_i)}{P_{q}(\lambda_i)}, \quad j\in [1,q]_{{\scriptscriptstyle\mathbb{Z}}}, \end{align*}$$
which yields
$$\begin{align*}\frac{P_{q-j}(\lambda_i)}{P_{q}(\lambda_i)}=\sum_{s=0}^j\alpha_{s,j}\lambda_i^s,\quad \alpha_{j,j}=A_qA_{q-1}\cdots A_{q-j+1}>0, \end{align*}$$
where the coefficients
$\alpha _{s,j}$
are independent of
$\lambda _i$
. Substituting these relations into (4.5), we get
From the interlacing of zeros of the polynomials
$P_q$
and
$\widetilde {P}_{q+1}$
, we have
$\operatorname {sign} P_q(\lambda _i)=(-1)^{i-1}$
,
$$\begin{align*}\operatorname{sign} \Delta(\lambda_1,\dots,\lambda_{m+1})=\operatorname{sign}{}(P_{q}(\lambda_1)\cdots P_{q}(\lambda_{m+1}))=(-1)^{\frac{m(m+1)}{2}}, \end{align*}$$
and
with
$c(q,m)>0$
.
By Lemma 4.1,
$D_{m+1}(\nu ,q-m+1,\dots ,q)$
preserves the sign for
$\nu \in [0,q-m]_{{\scriptscriptstyle \mathbb {Z}}}$
and then, according to (4.6), all
$a_{\nu }$
have the same sign. Further, since the polynomial (1.15) is positive for sufficiently large
$\lambda $
, it follows that
$a_{q-m}>0$
and thus all
$a_{\nu }>0$
. In particular, we have
$D_{m+1}(0,q-m+1,\dots ,q)>0$
, which yields, by Lemma 4.1,
$$ \begin{align} D_{m+1}(\nu,\nu_1,\dots,\nu_m)>0\quad \text{for all sequences}\quad 0\le \nu<\nu_1<\dots<\nu_m\le q. \end{align} $$
To show (1.16), let us write the difference of coefficients
$a_{\nu }$
in (1.15) using (4.6):
Applying formula (4.3) with
$\lambda =b$
and
$\widetilde {P}_{q+1}(b)=P_{q}(b)(\eta _{b}-\eta )$
, and also the Christoffel-Darboux formula with
$x=b$
and
$y=\lambda _i$
, we derive
Letting
$$\begin{align*}C_1=\frac{\eta_{b}-\eta}{b-\lambda_1},\quad C_2=\frac{P_{\nu}(b)P_{\nu+1}(b)P_{q}(b)w_{\nu}\prod_{j=q-m+1}^{q}w_j} {c(q,m)\prod_{j=2}^{m+1}(b-\lambda_j)\prod_{j=q-m+2}^{q}d_j}, \end{align*}$$
for
$\nu \in [0,q-m-1]_{{\scriptscriptstyle \mathbb {Z}}}$
, we have
$$ \begin{align} C_1 C_2 J_{\nu}= \sum_{\nu_2=\nu+1}^{q-m+1}\sum_{\nu_1=0}^{\nu}d_{\nu_1}d_{\nu_2}P_{\nu_1}(b)P_{\nu_2}(b)D_{m+1}(\nu_1,\nu_2,q-m+2,\dots,q). \end{align} $$
Moreover, for
$m=1$
,
$D_{2}(\nu _1,\nu _2,q-m+2,\dots ,q)=D_{2}(\nu _1,\nu _2)$
.
We note that the right-hand side of (4.8) (due to (4.7)) as well as the factor
$C_2$
are positive. We have already mentioned that, for
$\eta \neq \eta _b$
, the factor
$C_1$
is also positive, therefore
$J_{\nu }>0$
and (1.16) holds for
$m\ge 1$
.
(a): the case
$m\ge 1$
and
$\eta =\eta _b$
. Since, according to [Reference Szegö35, Ch. III, § 3.3], the function
$f(\lambda )=P_{q+1}(\lambda )/P_{q}(\lambda )$
on the interval
$(\lambda _{1,q},\infty )$
increases and
$f'(b)>0$
, by the inverse function rule,
$$\begin{align*}\lim_{\eta\to \eta_b}\frac{b-\lambda_1(\eta)}{\eta_b-\eta}=\frac{1}{f'(b)}>0. \end{align*}$$
Thus,
$C_1>0$
in (4.8) and (1.16) is also valid for
$\eta =\eta _b$
.
The proof of Theorem 1.14 is now complete.
5 Proof of Theorem 1.16
As proven in [Reference Ivanov20], polynomials (1.20) and (1.22) are the only extremal polynomials, up to a positive constant, in a version of Yudin’s problem, where the polynomial coefficients are not restricted to be non-negative. Thus, [Reference Ivanov20] provides the upper bounds for
$B_n(\mu ,m)$
in (1.19) and (1.21). To prove the lower bounds for
$B_n(\mu ,m)$
, it remains to check that polynomials (1.20) and (1.22) belong to
$\Pi _n\cap \Pi _{+} (\{U_l\})$
.
(a) In this case
$n=2q-m+1$
. According to Theorem 1.14 with
$\eta =0$
, we have
$$\begin{align*}\frac{U_{q+1}(t)}{(t-t_{1})\cdots(t-t_{m+1})}\in \Pi_{+}(\{U_l\}). \end{align*}$$
Applying the Krein property, we note that
$$\begin{align*}\frac{U_{q+1}^{2}(t)}{(t-t_{1})\cdots(t-t_{m+1})}\in \Pi_n\cap\Pi_{+}(\{U_l\}). \end{align*}$$
(b): the case
$n=2q-m+2$
and the polynomials
$U_{l}^{(1)}(t)$
satisfy the Krein condition. Since (see [Reference Levenshtein25])
$$\begin{align*}U_{l}^{(1)}(t)=\frac{\theta_lU_l(t)+(2-\theta_l)U_{l+1}(t)}{1+t},\quad \theta_l\in (0,2), \end{align*}$$
applying Theorem 1.14 for
$U_{q+1}^{(1)}(t)$
with
$\eta =0$
, we derive that the polynomial
$$ \begin{align*} \frac{(1+t)(U_{q+1}^{(1)}(t))^2}{(t-t_{1}^{(1)})\cdots (t-t_{m+1}^{(1)})}&=\sum_{l=0}^{q-m}d_l^{(1)}b_l(1+t)U_{q+1}^{(1)}(t)U_{l}^{(1)}(t) \\ &=\sum_{l=0}^{q-m}d_l^{(1)}b_l\sum_{k=q+1-l}^{q+1+l}c_{q+1,l,k}(1+t)U_{k}^{(1)}(t)\\ & = \sum_{l=0}^{q-m}d_l^{(1)}b_l\sum_{k=q+1-l}^{q+1+l}c_{q+1,l,k}(\theta_kU_k(t){+}(2-\theta_k)U_{k+1}(t)) \end{align*} $$
belongs to
$\Pi _n\cap \Pi _{+}(\{U_l\})$
.
(b): the case
$n=2q-m+2$
and
$\mu (t)$
be an odd function. Then the measure
$d\mu (t)$
is even,
$U_l(-1)=(-1)^l$
,
$\theta _l=1$
, and
$$ \begin{align} U_{l}^{(1)}(t)=\frac{U_{l}(t)+U_{l+1}(t)}{1+t}. \end{align} $$
Hence, according to the Christoffel-Darboux formula (2.1),
$$ \begin{align} U_{l}^{(1)}(t)=w_l^{-1}\sum_{\nu=0}^{l}d_{\nu}(-1)^{l+\nu}U_{\nu}(t). \end{align} $$
Applying (5.1) and (5.2), we get
$$\begin{align*}\int_{-1}^{1}(U_{l}^{(1)}(t))^2\,(1+t)\,d\mu(t)= \int_{-1}^{1}w_l^{-1}\sum_{\nu=0}^{l}d_{\nu}(-1)^{l+\nu}U_{\nu}(t)(U_{l}(t)+U_{l+1}(t))\,d\mu(t)=w_l^{-1}. \end{align*}$$
This, Theorem 1.14, and (5.2) imply that in the expansion
$$\begin{align*}\frac{U_{q+1}^{(1)}(t)}{(t-t_{1}^{(1)})\cdots (t-t_{m+1}^{(1)})}= \sum_{l=0}^{q-m}w_lb_{l}U_{l}^{(1)}(t) \end{align*}$$
the coefficients are monotone, that is,
$b_0>b_1>\dots >b_{q-m}>0$
. Applying (5.2), we obtain
$$\begin{align*}\sum_{l=0}^{q-m}w_lb_{l}U_{l}^{(1)}(t)=\sum_{l=0}^{q-m}b_{l}\sum_{\nu=0}^{l}d_{\nu}(-1)^{l+\nu}U_{\nu}(t) =\sum_{\nu=0}^{q-m}d_{\nu}U_{\nu}(t)\sum_{l=\nu}^{q-m}b_{l}(-1)^{l+\nu}=\sum_{\nu=0}^{q-m}d_{\nu}\delta_{\nu}U_{\nu}(t), \end{align*}$$
where
$$\begin{align*}\delta_{\nu}=\sum_{l=\nu}^{q-m}b_{l}(-1)^{l+\nu}=b_{\nu}-b_{\nu+1}+\dots+(-1)^{\nu+q-m}b_{q-m}\ge 0. \end{align*}$$
Thus,
$$\begin{align*}\frac{(1+t)(U_{q+1}^{(1)}(t))^2}{(t-t_{1}^{(1)})\cdots(t-t_{m+1}^{(1)})}= \sum_{\nu=0}^{q-m}d_{\nu}\delta_{\nu}U_{\nu}(t)(U_{q+1}(t)+U_{q+2}(t))\in \Pi_n\cap\Pi_{+}(\{U_l\}), \end{align*}$$
completing the proof.
Appendix
A.1 Calculation of determinants for Chebyshev polynomials
In this subsection, we calculate the determinants
$D(\nu )$
(see (4.1)) for the Jacobi polynomials [Reference Szegö35, Ch. IV, § 4.1] given by
$$\begin{align*}U_l^{(\alpha,\beta)}(t)=\frac{P_{l}^{(-\frac{1}{2}+\alpha,-\frac{1}{2}+\beta)}(t)} {P_{l}^{(-\frac{1}{2}+\alpha,-\frac{1}{2}+\beta)}(1)},\quad \alpha,\beta\in\{0,1\}. \end{align*}$$
It will be convenient to rearrange
$\nu _i$
as follows:
$1\le \nu _{m}<\dots <\nu _{1}\le q$
.
(i) If
$\alpha =\beta =0$
,
$U_{q+1}^{(0,0)}(t)=P_{q+1}^{(-\frac {1}{2},-\frac {1}{2})}(t)=\cos {}((q+1)\arccos t)$
is the Chebyshev polynomial of the first kind and
$t_j=\cos \frac {\pi (2j-1)}{2q+2}$
,
$j=1,\dots ,q+1$
, are its zeros. If
$x_i=\frac {\pi \nu _i}{2q+2}$
, then
$$\begin{align*}D(\nu)=D(\nu,\nu_m,\dots,\nu_1)=\det{}(\cos{}(2j-1)x_i)_{i,j=1}^{m+1}. \end{align*}$$
Since
$\cos {}(2j-1)x_i=\sum _{s=0}^{j-1}c_{s}\cos ^{2s+1}x_i$
,
$c_{j-1}= 2^{2j-2}$
, the determinant
$D(\nu )$
reduces to the Vandermonde determinant
$$ \begin{align*} D(\nu)&=\det{}(2^{2j-2}(\cos x_i)^{2j-1})_{i,j=1}^{m+1}=2^{m(m+1)}\prod_{i=1}^{m+1}(\cos x_i)\det{}((\cos^{2}x_i)^{j-1})_{i,j=1}^{m+1}\\ &=2^{m(m+1)}\prod_{i=1}^{m+1}(\cos x_i)\prod_{1\le k<l\le m+1}(\cos^2x_l-\cos^2x_k)\\ &=2^{m(m+1)}\prod_{i=1}^{m+1}(\cos x_i)\prod_{1\le k<l\le m}(\cos^2x_l-\cos^2x_k)\,\prod_{1\le k\le m}\Big(\cos^2\Big(\frac{\pi\nu}{2q+2}\Big)-\cos^2\Big(\frac{\pi\nu_k}{2q+2}\Big)\Big). \end{align*} $$
Taking into account that
$0<\cos x_1<\dots <\cos x_{m+1}<1$
, we obtain
$$\begin{align*}\operatorname{sign} D(\nu)=\operatorname{sign} \prod_{k=1}^m \Big(\cos\Big(\frac{\pi\nu}{2q+2}\Big)-\cos\Big(\frac{\pi\nu_k}{2q+2}\Big)\Big)=\operatorname{sign} \prod_{k=1}^m(\nu_k-\nu). \end{align*}$$
(ii) The polynomial
$U_{q+1}^{(0,1)}(t)$
is given by
$$\begin{align*}U_{q+1}^{(0,1)}(t)=P_{q+1}^{(-\frac{1}{2},\frac{1}{2})}(t)=\frac{\sqrt{2}\cos{}((q+3/2)\arccos t)} {\sqrt{1+t}} \end{align*}$$
and
$t_j=\cos \frac {\pi (2j-1)}{2q+3}$
,
$j=1,\dots ,q+1$
, are its zeros. If
$x_i=\frac {\pi (\nu _i+1/2)}{2q+3}$
, then as above
$$ \begin{align*} D(\nu)&=\det \Big(\frac{\sqrt{2}\cos{}(2j-1)x_i}{\sqrt{1+t_j}}\Big)_{i,j=1}^{m+1}=\prod_{j=1}^{m+1}\Big(\frac{2}{1+t_j}\Big)^{1/2} \det \bigl(\cos{}(2j-1)x_i\bigr)_{i,j=1}^{m+1}\\ &=2^{m(m+1)}\prod_{j=1}^{m+1}\Big(\frac{2\cos^2 x_j}{1+t_j}\Big)^{1/2}\,\prod_{0\le k<l\le m+1}(\cos^2x_l-\cos^2x_k). \end{align*} $$
(iii) The polynomial
$$\begin{align*}U_{q+1}^{(1,1)}(t)=P_{q+1}^{(\frac{1}{2},\frac{1}{2})}(t)=\frac{\sin((q+2)\arccos t)} {(q+2)\sqrt{1-t^2}}, \end{align*}$$
is the Chebyshev polynomial of the second kind with the zeros
$t_j=\cos \frac {\pi j}{q+2}$
,
$j=1,\dots ,q+1$
. If
$x_i=\frac {\pi (\nu _i+1)}{q+2}$
, then
$$\begin{align*}D(\nu)=\det \Big(\frac{\sin jx_i}{(\nu_i+1)\sin\frac{\pi j}{q+2}}\Big)_{i,j=1}^{m+1}= \frac{\det{}(\sin jx_i)_{i,j=1}^{m+1}}{\prod_{j=1}^{m+1}(\nu_j+1)\sin\frac{\pi j}{q+2}}. \end{align*}$$
Since
$\sin jx_i=\sin x_i\,\sum _{s=0}^{j-1}c_{s}\cos ^{s}x_i$
with
$c_{j-1}=2^{j -1}$
, we arrive at
$$ \begin{align*} D(\nu)&=2^{\frac{m(m+1)}{2}}\prod_{j=1}^{m+1}\frac{\sin\frac{\pi(\nu_j+1)}{q+2}}{(\nu_j+1)\sin\frac{\pi j}{q+2}}\,\det{}(\cos^{j-1}x_i)_{i,j=1}^{m+1}\\ &=2^{\frac{m(m+1)}{2}}\prod_{j=1}^{m+1}\frac{\sin\frac{\pi(\nu_j+1)}{q+2}}{(\nu_j+1)\sin\frac{\pi j}{q+2}}\, \prod_{0\le k<l\le m+1}(\cos x_l-\cos x_k). \end{align*} $$
(iv) We have
$$\begin{align*}U_{q+1}^{(1,0)}(t)=P_{q+1}^{(\frac{1}{2},- \frac{1}{2})}=\frac{\sqrt{2}\sin((q+3/2)\arccos t)} {(2q+3)\sqrt{1-t}}, \end{align*}$$
and
$t_j=\cos \frac {\pi j}{q+3/2}$
,
$j=1,\dots ,q+1$
, are the zeros of
$U_{q+1}^{(1,0)}$
. As in the previous case, for
$x_i=\frac {\pi (\nu _i+1/2)}{q+3/2}$
,
$$ \begin{align*} D(\nu)&=\det \Big(\frac{\sqrt{2}\sin jx_i}{(2\nu_i+1)\sqrt{1-t_j}}\Big)_{i,j=1}^{m+1}= \frac{\det{}(\sin jx_i)_{i,j=1}^{m+1}}{\prod_{j=1}^{m+1}(2\nu_j+1)\sqrt{(1-t_j)/2}}\\ &=2^{\frac{m(m+1)}{2}}\prod_{j=1}^{m+1}\frac{\sin\frac{\pi(\nu_j+1)}{q+2}}{(2\nu_j+1)\sqrt{(1-t_j)/2}}\prod_{0\le k<l\le m+1}(\cos x_l-\cos x_k). \end{align*} $$
A.2 Corollaries of Theorem 1.14 for trigonometric polynomials
Finally, writing the Chebyshev polynomials
$U_{q+1}^{(0,0)}$
and
$U_{q+1}^{(1,1)}$
in the trigonometric form, the monotonicity property of coefficients in expansion (1.15) given in Theorem 1.14 implies the following results.
Corollary A.1. Suppose
$0\le m\le q$
, then
$$ \begin{align} \frac{\cos{}(q+1)x}{\bigl(\cos x-\cos\frac{\pi}{2q+2}\bigr)\cdots\bigl(\cos x-\cos\frac{\pi(2m+1)}{2q+2}\bigr)}= \frac{a_{0,m}}{2}+\sum_{\nu=1}^{q-m}a_{\nu,m}\cos\nu x, \end{align} $$
then
$$\begin{align*}a_{0,m}>a_{1,m}>\dots>a_{q-m,m}>0. \end{align*}$$
Corollary A.2. Suppose
$1\le m\le q$
, then
$$\begin{align*}\frac{\sin(q+1)x}{\bigl(\cos x-\cos\frac{\pi}{q+1}\bigr)\cdots\bigl(\cos x-\cos\frac{\pi m}{q+1}\bigr)}= \sum_{\nu=1}^{q-m+1}\nu b_{\nu,m}\sin\nu x, \end{align*}$$
then
$$\begin{align*}b_{1,m}>\dots>b_{q-m+1,m}>0. \end{align*}$$
It is worth mentioning that polynomials similar to (A.1) were considered by Yudin in [Reference Yudin40].
Acknowledgements
We would like to thank the anonymous reviewers for their valuable and constructive comments, which allowed us to significantly improve the article.
Competing interest
The authors have no competing interests to declare.
Funding statement
The proofs of Theorems 1.4, 1.8, 1.10, 1.14 were supported by the Russian Science Foundation (project no. 23-71-30001) at the Lomonosov Moscow State University. The work of the third author was partially supported by grants PID2023-150984NB-I00, 2021 SGR 00087, the CERCA Programme of the Generalitat de Catalunya, and by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R
$\&$
D (CEX2020-001084-M).
Author contributions
All authors contributed equally to this work.
Open access
