1.1 Cyclic singular inner functions

Let X be a topological space consisting of functions which are analytic in the unit disk $\mathbb {D} = \{ z \in \mathbb {C} : |z| < 1\}$ and which satisfy some customary desirable properties, such as that the evaluation $f \mapsto f(\lambda )$ is a continuous functional on X for each $\lambda \in \mathbb {D}$ and that the function $z \mapsto zf(z)$ is contained in the space X whenever $f \in X$ . A function $g \in X$ is said to be cyclic if there exists a sequence of analytic polynomials $\{p_n\}_n$ for which the polynomial multiples $\{gp_n\}_n$ converge to the constant function $1$ in the topology of the space.

The well-known Hardy classes $H^p$ are among the very few examples of analytic function spaces in which the cyclicity phenomenon is completely understood. The cyclic functions g are of the form

(1.1) $$ \begin{align} g(z) = \exp\Bigg( \int_{\mathbb{T}} \frac{\zeta + z}{\zeta - z} \log(|g(\zeta)|) \, d\textit{m}(\zeta)\Bigg), \quad z \in \mathbb{D}, \end{align} $$

where $d\textit {m}$ is the (normalized) Lebesgue measure of the unit circle $\mathbb {T} = \{ z\in \mathbb {C} : |z| = 1\}$ . Functions as in (1.1) are called outer functions. The inner functions are of the form

(1.2) $$ \begin{align} \theta(z) & = B(z) S_\nu(z) \nonumber \\ & = \prod_{n} \frac{\overline{\alpha_n}}{|\alpha_n|}\frac{\alpha_n - z}{1-\overline{\alpha_n}z} \cdot \exp\Bigg( -\int_{\mathbb{T}} \frac{\zeta + z}{\zeta - z} d\nu(\zeta)\Bigg), \quad z \in \mathbb{D}, \end{align} $$

where $\nu $ is a positive finite singular Borel measure on $\mathbb {T}$ and $\{\alpha _n\}_n$ is a Blaschke sequence. It is clear that if the Blaschke product B on the left is nontrivial, then $\theta $ vanishes at points in $\mathbb {D}$ and therefore cannot be cyclic in any reasonable space of analytic functions X. The right factor $S_\nu $ is a singular inner function, and it is well known that if a function $g \in H^p$ has a singular inner function as a factor, then g is not cyclic in $H^p$ . As a consequence, if $\{p_n\}_n$ is a sequence of polynomials for which we have

$$ \begin{align*}\lim_{n \to \infty} \theta(z)p_n(z) = 1, \quad z \in \mathbb{D},\end{align*} $$

then necessarily the Hardy class norms of the sequence must explode

$$ \begin{align*}\lim_{n \to \infty} \, \|\theta p_n\|^p_{H^p} := \lim_{n \to \infty} \int_{\mathbb{T}} |\theta p_n|^p d\textit{m} = \infty\end{align*} $$

for finite $p \geq 1$ , or in case $p = \infty $ ,

$$ \begin{align*}\lim_{n \to \infty} \|\theta p_n\|_\infty := \lim_{n \to \infty}\sup_{z \in \mathbb{D}} |\theta(z)p_n(z)| = \infty.\end{align*} $$

When other norms are considered, cyclic singular inner functions might exist, and here the Bergman spaces $L^p_a(\mathbb {D})$ provide a famous set of examples. The Bergman norms are of the form

$$ \begin{align*}\|g\|^p_{L^p(\mathbb{D})} := \int_{\mathbb{D}} |g(z)|^p dA(z),\end{align*} $$

where $dA$ is the normalized area measure of $\mathbb {D}$ . After a sequence of partial results by multiple authors, Korenblum in [Reference Korenblum12] and Roberts in [Reference Roberts15] independently characterized the cyclic singular inner functions in the Bergman spaces in terms of the vanishing on certain subsets of $\mathbb {T}$ of the corresponding singular measure $\nu $ appearing in (1.2). A construction of a singular inner function which is cyclic in the classical Bloch space appears in [Reference Anderson, Fernandez and Shields3].

Recently, Ransford in [Reference Ransford14] noted that singular inner functions exist which decay arbitrarily slowly near the boundary of the disk. As we shall see below, this fact has as a direct consequence the existence of an abundance of spaces of analytic function which admit cyclic singular inner functions. Here is the precise statement of the main result of [Reference Ransford14].

Theorem 1.1 Let $w:[0,1) \to (0,1)$ be any function satisfying $\lim _{r \to 1^-} w(r) = 0$ . Then there exists a singular inner function $\theta $ for which we have

(1.3) $$ \begin{align} \min_{|z| < r} |\theta(z)| \geq w(r), \quad r \in (0,1). \end{align} $$

It has been remarked to the present author that, in fact, this theorem appears already in the literature. For instance, Shapiro similarly mentions in [Reference Shapiro17] that a singular inner function always satisfies an estimate of the form

(1.4) $$ \begin{align} |S_\nu(z)| \geq \exp\Bigg(- C \frac{\omega(1-|z|)}{1-|z|}\Bigg), \end{align} $$

where C is some positive constant, and $\omega = \omega _\nu $ is the modulus of continuity of the measure $\nu $ :

(1.5) $$ \begin{align} \omega_\nu(h) = \sup_{|I| = h} \nu(I). \end{align} $$

The supremum above is taken over arcs I of the circle $\mathbb {T}$ which are of length h. In [Reference Shapiro18], Shapiro proves that a singular measure $\nu $ exists with a modulus of continuity $\omega _\nu $ for which $\omega _\nu (h)/h$ grows to infinity arbitrarily slowly as h decreases to zero, hence proving Theorem 1.1 as a consequence of the estimate (1.4). In fact, such singular measures have been known to exist at least since the work of Hartman and Kershner in [Reference Hartman and Kershner10]. The proof of Ransford in [Reference Ransford14] also involves establishing the existence of such a measure.

The following result is the abovementioned consequence of Theorem 1.1 on existence of cyclic singular inner function. The result is surely well known, and has an elementary proof which we include for convenience.

Proof Apply Theorem 1.1 to the function w to produce a singular inner function $\theta $ satisfying (1.3). For integers $n \geq 2$ , we set $r_n := 1- 1/n$ and $Q_n(z) := 1/\theta (r_n z)$ . Then $Q_n$ is holomorphic in a neighborhood of the closed disk $\overline {\mathbb {D}}$ , and because we are assuming that w is decreasing, we have the estimate

$$ \begin{align*}\sup_{z \in \mathbb{D}} |Q_n(z)|w(|z|) \leq \sup_{z \in \mathbb{D}} \frac{w(|z|)}{w(r_n|z|)} \leq 1.\end{align*} $$

We can approximate $Q_n$ by an analytic polynomial $p_n$ so that

$$ \begin{align*}\sup_{z \in \mathbb{D}} |Q_n(z) - p_n(z)| \leq 1/n.\end{align*} $$

Then

$$ \begin{align*} \sup_{z \in \mathbb{D}} |\theta(z) p_n(z)| w(|z|) \leq \sup_{z \in \mathbb{D}} \Big(|\theta(z) Q_n(z)| + 1/n \Big) w(|z|) \leq 2. \end{align*} $$

It is clear from the construction that $\theta (z)p_n(z) \to 1$ as $n \to \infty $ , for any $z \in \mathbb {D}$ .

Corollary 1.2 says that there exist cyclic singular inner functions in essentially any space of analytic functions defined in terms of a growth condition, or in any space in which such a growth space is continuously embedded.

The purpose of this note is to apply Theorem 1.1, or more precisely its simple consequence stated in Corollary 1.2, to the questions of existence of functions with certain smoothness properties in model spaces $K_\theta $ . We will establish sharpness of certain existing approximation results in these spaces. Moreover, we take the opportunity to discuss similar questions in the broader class of de Branges–Rovnyak spaces $\mathcal {H}(b)$ . Our results are proved by rather well-known methods, but their statements seem to be missing in the existing literature, and we wish to fill in this gap.

In the proofs of the main results, which will be stated shortly, we will concern ourselves with the following weak type of cyclicity of singular inner functions. Let Y be some linear space of analytic functions which is contained in $H^1$ . We want to investigate if there exist a singular inner function $\theta $ and a sequence of polynomials $\{p_n\}_n$ such that

(1.6) $$ \begin{align} f(0) = \int_{\mathbb{T}} f \, d\textit{m} = \lim_{n \to \infty} \int f \overline{\theta p_n} \,d\textit{m} \end{align} $$

holds for all $f \in Y$ . The above situation means that the sequence $\{\theta p_n\}_n$ converges to the constant $1$ , weakly over the space Y. Now, clearly, if Y is too large of a space (say, $Y = H^2$ ), then (1.6) can never hold for all $f \in Y$ . However, if Y is sufficiently small, then the situation in (1.6) might occur. For instance, in the extreme case, when Y is a set of analytic polynomials, then any singular inner function $\theta $ and any sequence of polynomials $\{p_n\}_n$ which satisfies $\lim _{n \to \infty } p_n(z) = 1/\theta (z)$ for $z \in \mathbb {D}$ is sufficient to make (1.6) hold. Philosophically speaking, it is the uniform smoothness of the functions in the class Y that allows the existence of singular inner functions $\theta $ for which the above situation occurs. Under insignificant assumptions on Y, a straightforward argument shows that if (1.6) occurs, then the intersection between Y and $K_\theta $ is trivial, whereas Corollary 1.2 provides us with a huge class of spaces Y for which (1.6) can be achieved.

1.2 Main results

Recall that the space $K_\theta $ is constructed from an inner function $\theta $ by taking the orthogonal complement of the subspace

$$ \begin{align*}\theta H^2 := \{ \theta h : h \in H^2\}\end{align*} $$

in the Hardy space $H^2$ :

$$ \begin{align*} K_\theta = H^2 \ominus \theta H^2. \end{align*} $$

For background on the spaces $K_\theta $ , one can consult the books [Reference Cima, Matheson and Ross5, Reference Garcia, Mashreghi and Ross9]. In our first result, we will show that the famous approximation theorem of Aleksandrov from [Reference Aleksandrov1] on density in $K_\theta $ of functions which extend continuously to the boundary is in fact essentially sharp, as it cannot be extended to any class of functions satisfying an estimate on their modulus of continuity. By a modulus of continuity $\omega $ , we mean here a function $\omega : [0,\infty ) \to [0, \infty )$ which is continuous, increasing, satisfies $\omega (0) = 0$ , and for which $\omega (t)/t$ is a decreasing function with

$$ \begin{align*}\lim_{t \to 0^+} \omega(t)/t = \infty.\end{align*} $$

For such a function $\omega $ , we define $\Lambda ^\omega _a$ to be the space of functions f which are analytic in $\mathbb {D}$ , extend continuously to $\overline {\mathbb {D}}$ , and satisfy

(1.7) $$ \begin{align} \sup_{z,w \in \overline{\mathbb{D}}, z \neq w} \frac{|f(z)-f(w)|}{\omega(|z-w|)} < \infty. \end{align} $$

Then $\Lambda ^\omega _a$ is the space of analytic functions on $\mathbb {D}$ which have a modulus of continuity dominated by $\omega $ . We make $\Lambda ^\omega _a$ into a normed space by introducing the quantity

$$ \begin{align*}\|f\|_\omega := \|f\|_\infty + \sup_{z,w \in \overline{\mathbb{D}}, z \neq w} \frac{|f(z)-f(w)|}{\omega(|z-w|)}.\end{align*} $$

By a theorem of Tamrazov from [Reference Tamrazov19], we could have replaced the supremum over $\overline {\mathbb {D}}$ by a supremum over $\mathbb {T}$ , and obtain the same space of functions (we remark that a nice proof of this result is contained in [Reference Bouya4, Appendix A]). The following is an optimality statement regarding Aleksandrov’s density theorem.

Theorem 1.3 Let $\omega $ be a modulus of continuity. There exists a singular inner function $\theta $ such that

$$ \begin{align*}\Lambda^\omega_a \cap K_\theta = \{0\}.\end{align*} $$

This statement will be proved in Section 3. In fact, we will see that Theorem 1.3 is a consequence of a variant, and in some directions a strengthening, of a theorem of Dyakonov and Khavinson from [Reference Dyakonov and Khavinson6]. For a sequence of positive numbers $\boldsymbol {\lambda } = \{\lambda _n\}_{n=0}^\infty $ , we define the class

(1.8) $$ \begin{align} H^2_{\boldsymbol{\lambda}} = \Bigg\{ f = \sum_{n=0}^\infty f_n z^n \in \mathrm{Hol}(\mathbb{D}) : \sum_{n=0}^\infty \lambda_n|f_n|^2 < \infty \Bigg\}. \end{align} $$

The next theorem, proved in Section 2, reads as follows.

Theorem 1.4 Let $\boldsymbol {\lambda } = \{\lambda _n\}_{n=0}^\infty $ be any increasing sequence of positive numbers with $\lim _{n \to \infty } \lambda _n = \infty $ . Then there exists a singular inner function $\theta $ such that

$$ \begin{align*}H^2_{\boldsymbol{\lambda}} \cap K_\theta = \{0\}.\end{align*} $$

The result can be compared to the mentioned result of Dyakonov and Khavinson in [Reference Dyakonov and Khavinson6], from which the above result can be deduced in the special case $\boldsymbol {\lambda } = \{ (k+1)^{\alpha }\}_{k=0}^\infty $ with any $\alpha> 0$ .

The theory of de Branges–Rovnyak spaces $\mathcal {H}(b)$ is a well-known generalization of the theory of model spaces $K_\theta $ . The symbol of the space b is now any analytic self-map of the unit disk, and we have $\mathcal {H}(b) = K_b$ whenever b is inner. For background on $\mathcal {H}(b)$ spaces, one can consult [Reference Sarason16] or [Reference Fricain and Mashreghi7, Reference Fricain and Mashreghi8]. A consequence of the author’s work in collaboration with Aleman in [Reference Aleman and Malman2] is that the abovementioned density theorem of Aleksandrov generalizes to the broader class of $\mathcal {H}(b)$ spaces: any such space admits a dense subset of functions which extend continuously to the boundary. Since Theorem 1.3 proves optimality of Aleksandrov’s theorem for inner functions $\theta $ , one could ask if at least for outer symbols b any improvement of the density result in $\mathcal {H}(b)$ from [Reference Aleman and Malman2] can be obtained. In Section 4, we remark that this is not the case, and the result in [Reference Aleman and Malman2] is also essentially optimal, even for outer symbols b.

Theorem 1.5 Let $\boldsymbol {\lambda } = \{\lambda _n\}_{n=0}^\infty $ be any increasing sequence of positive numbers with $\lim _{n \to \infty } \lambda _n = \infty $ . There exists an outer function $b:\mathbb {D} \to \mathbb {D}$ such that

$$ \begin{align*}H^2_{\boldsymbol{\lambda}} \cap \mathcal{H}(b)= \{0\}.\end{align*} $$

Theorem 1.6 Let $\omega $ be a modulus of continuity. There exists an outer function $b:\mathbb {D} \to \mathbb {D}$ such that

$$ \begin{align*}\Lambda^\omega_a \cap \mathcal{H}(b) = \{ 0 \}.\end{align*} $$

We will show that the above results are essentially equivalent to a theorem of Khrushchev from [Reference Khrushchev11].

In Section 5, we list a few questions we have not found an answer for, and some ideas for further research.