We study approximations for the Lévy area of Brownian motion which are based on the Fourier series expansion and a polynomial expansion of the associated Brownian bridge. Comparing the asymptotic convergence rates of the Lévy area approximations, we see that the approximation resulting from the polynomial expansion of the Brownian bridge is more accurate than the Kloeden–Platen–Wright approximation, whilst still only using independent normal random vectors. We then link the asymptotic convergence rates of these approximations to the limiting fluctuations for the corresponding series expansions of the Brownian bridge. Moreover, and of interest in its own right, the analysis we use to identify the fluctuation processes for the Karhunen–Loève and Fourier series expansions of the Brownian bridge is extended to give a stand-alone derivation of the values of the Riemann zeta function at even positive integers.

Given a sequence \varrho =(r_n)_n\in [0,1)$\varrho =(r_n)_n\in [0,1)$ tending to 1$1$, we consider the set {\mathcal {U}}_A({\mathbb {D}},\varrho )${\mathcal {U}}_A({\mathbb {D}},\varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk \mathbb {D}$\mathbb {D}$ such that for any compact set K included in the unit circle {\mathbb {T}}${\mathbb {T}}$, different from {\mathbb {T}}${\mathbb {T}}$, the set \{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$\{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$ is dense in the space {\mathcal {C}}(K)${\mathcal {C}}(K)$ of continuous functions on K. It is known that the set {\mathcal {U}}_A({\mathbb {D}},\varrho )${\mathcal {U}}_A({\mathbb {D}},\varrho )$ is residual in H(\mathbb {D})$H(\mathbb {D})$. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at 0$0$ are dense in {\mathcal {C}}(K)${\mathcal {C}}(K)$ for any compact set K\subset {\mathbb {T}}$K\subset {\mathbb {T}}$ different from {\mathbb {T}}${\mathbb {T}}$. Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators T_n:f\mapsto f(r_n \cdot )$T_n:f\mapsto f(r_n \cdot )$ acting on H(\mathbb {D})$H(\mathbb {D})$. Thus, we study the dynamical properties of (T_n)_n$(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.

If the logarithmic dimension of a Cantor-type set K is smaller than 1$1$ , then the Whitney space \mathcal {E}(K)$\mathcal {E}(K)$ possesses an interpolating Faber basis. For any generalized Cantor-type set K, a basis in \mathcal {E}(K)$\mathcal {E}(K)$ can be presented by means of functions that are polynomials locally. This gives a plenty of bases in each space \mathcal {E}(K)$\mathcal {E}(K)$ . We show that these bases are quasi-equivalent.

This paper studies a new Whitney type inequality on a compact domain \Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$ that takes the form

\begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*} $$\begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0<p\leq \infty, \end{align*}$$

\omega _{\mathcal {E}}^r(f, t)_p$\omega _{\mathcal {E}}^r(f, t)_p$

f\in L^p(\Omega )$f\in L^p(\Omega )$

\mathcal {E}\subset \mathbb {S}^{d-1}$\mathcal {E}\subset \mathbb {S}^{d-1}$

\mathrm {span}(\mathcal {E})={\mathbb R}^d$\mathrm {span}(\mathcal {E})={\mathbb R}^d$

\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$

\mathcal {E}$\mathcal {E}$

\Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$

C^2$C^2$

\Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$

\mathcal {E}$\mathcal {E}$

\mathcal {N}_d(\Omega )\in {\mathbb N}$\mathcal {N}_d(\Omega )\in {\mathbb N}$

\mathcal {N}_d(\Omega )$\mathcal {N}_d(\Omega )$

\mathcal {E}$\mathcal {E}$

\mathrm {span}(\mathcal {E})={\mathbb R}^d$\mathrm {span}(\mathcal {E})={\mathbb R}^d$

\Omega $\Omega$

r\in {\mathbb N}$r\in {\mathbb N}$

p>0$p>0$

\mathcal {N}_d(\Omega )=d$\mathcal {N}_d(\Omega )=d$

C^2$C^2$

\Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$

d=2$d=2$

\Omega \subset {\mathbb R}^2$\Omega \subset {\mathbb R}^2$

d\ge 3$d\ge 3$

\Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$

d\ge 3$d\ge 3$

\Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$

\mathcal {N}_d(\Omega )$\mathcal {N}_d(\Omega )$

\Omega $\Omega$

\Omega $\Omega$

\mathcal {E}\subset \mathbb {S}^{d-1}$\mathcal {E}\subset \mathbb {S}^{d-1}$

\mathcal {E}$\mathcal {E}$

\Omega $\Omega$

r\in {\mathbb N}$r\in {\mathbb N}$

0<p\leq \infty $0<p\leq \infty$

\mathcal {N}_d(\Omega )$\mathcal {N}_d(\Omega )$

d\ge 3$d\ge 3$

A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body \Omega \subset {\mathbb R}^d$\Omega \subset {\mathbb R}^d$ , but with the set \mathcal {E}$\mathcal {E}$ containing more than (c d)^{d-1}$(c d)^{d-1}$ directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.

We study the metric projection onto the closed convex cone in a real Hilbert space \mathscr {H}$\mathscr {H}$ generated by a sequence \mathcal {V} = \{v_n\}_{n=0}^\infty $\mathcal {V} = \{v_n\}_{n=0}^\infty$ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by \mathcal {V}$\mathcal {V}$ coincides with the following set:

\begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$\begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*}$$

\mathcal {C}[[\mathcal {V}]]$\mathcal {C}[[\mathcal {V}]]$

L^2([-1,1])$L^2([-1,1])$

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in d$d$ dimensions, where d$d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the corresponding Nikolskii inequality for the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of d$d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.

We investigate convergence in the cone of completely monotone functions. Particular attention is paid to the approximation of and by exponentials and stretched exponentials. The need for such an analysis is a consequence of the fact that although stretched exponentials can be approximated by sums of exponentials, exponentials cannot in general be approximated by sums of stretched exponentials.

In this paper we consider the algorithm for recovering sparse orthogonal polynomials using stochastic collocation via ℓq minimization. The main results include: 1) By using the norm inequality between ℓq and ℓ2 and the square root lifting inequality, we present several theoretical estimates regarding the recoverability for both sparse and non-sparse signals via ℓq minimization; 2) We then combine this method with the stochastic collocation to identify the coefficients of sparse orthogonal polynomial expansions, stemming from the field of uncertainty quantification. We obtain recoverability results for both sparse polynomial functions and general non-sparse functions. We also present various numerical experiments to show the performance of the ℓq algorithm. We first present some benchmark tests to demonstrate the ability of ℓq minimization to recover exactly sparse signals, and then consider three classical analytical functions to show the advantage of this method over the standard ℓ1 and reweighted ℓ1 minimization. All the numerical results indicate that the ℓq method performs better than standard ℓ1 and reweighted ℓ1 minimization.

We introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain the hp-version bound on the numerical error of the multiple interval collocation method under H 1-norm. Numerical experiments confirm the theoretical expectations.

Various numerical methods have been developed in order to solve complex systems with uncertainties, and the stochastic collocation method using l1-minimisation on low discrepancy point sets is investigated here. Halton and Sobol' sequences are considered, and low discrepancy point sets and random points are compared. The tests discussed involve a given target function in polynomial form, high-dimensional functions and a random ODE model. Our numerical results show that the low discrepancy point sets perform as well or better than random sampling for stochastic collocation via l1-minimisation.

In this paper, we prove that for \ell \,=\,1$\ell \,=\,1$ or 2 the rate of best \ell $\ell$ - monotone polynomial approximation in the {{L}_{p}}${{L}_{p}}$ norm \left( 1\,\le \,p\,\le \,\infty \right)$\left( 1\,\le \,p\,\le \,\infty \right)$ weighted by the Jacobi weight {{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with \alpha ,\,\beta \,>\,-1/p$\alpha ,\,\beta \,>\,-1/p$ if p\,<\,\infty $p\,<\,\infty$ , or \alpha ,\,\beta \,\ge \,0$\alpha ,\,\beta \,\ge \,0$ if p\,=\,\infty $p\,=\,\infty$ , is bounded by an appropriate \left( \ell \,+\,1 \right)$\left( \ell \,+\,1 \right)$ -st modulus of smoothness with the same weight, and that this rate cannot be bounded by the \left( \ell \,+\,2 \right)$\left( \ell \,+\,2 \right)$ -nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

We consider the classical problem of finding the best uniform approximation by polynomials of 1/(x-a)^2,$1/(x-a)^2,$ where a>1$a>1$ is given, on the interval [-\! 1,1]$[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.

In this paper, we present and analyze a single interval Legendre-Gauss spectral collocation method for solving the second order nonlinear delay differential equations with variable delays. We also propose a novel algorithm for the single interval scheme and apply it to the multiple interval scheme for more efficient implementation. Numerical examples are provided to illustrate the high accuracy of the proposed methods.

In this paper we deal with the operators

\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray*} $$\begin{eqnarray*}{Z}_{n} (f; x)= \frac{n}{{b}_{n} } { \mathop{\sum }\nolimits}_{k= 0}^{n} {p}_{n, k} \biggl(\frac{x}{{b}_{n} } \biggr)\int \nolimits \nolimits_{0}^{\infty } {s}_{n, k} \biggl(\frac{t}{{b}_{n} } \biggr)f(t)\hspace{0.167em} dt, \quad 0\leq x\leq {b}_{n}\end{eqnarray*}$$

{p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1] ${p}_{n, k} (u)=\bigl(\hspace{-4pt}{\scriptsize \begin{array}{ l} \displaystyle n\\ \displaystyle k\end{array} } \hspace{-4pt}\bigr){u}^{k} \mathop{(1- u)}\nolimits ^{n- k} , (0\leq k\leq n), u\in [0, 1]$

{s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )${s}_{n, k} (u)= {e}^{- nu} \mathop{(nu)}\nolimits ^{k} \hspace{-3pt}/ k!, u\in [0, \infty )$

A unified process for the construction of hierarchical conforming bases on a range of element types is proposed based on an ab initio preservation of the underlying cohomology. This process supports not only the most common simplicial element types, as are now well known, but is generalized to squares, hexahedra, prisms and importantly pyramids. Whilst these latter cases have received (to varying degrees) attention in the literature, their foundation is less well developed than for the simplicial case. The generalization discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair and Arnold et al. for simplices. The process of forming hierarchical bases involves a recursive orthogonalization and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or better growth in condition number.

We propose some new weighted averaging methods for gradient recovery, and present analytical and numerical investigation on the performance of these weighted averaging methods. It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension. Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is analytic. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and feature non-uniform convergence. This characteristic behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

The stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional (random) parameter space. In addition to providing highly accurate stochastic solutions, the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters. In this paper, we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol’. This method allows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids. We discuss convergence of this method, apply it to several test cases and compare to existing methods. As a result which may be of independent interest, we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema. This allows one to manipulate the sparse grid collocation results in a highly efficient manner.

We construct bivariate polynomial approximations of the Lerch function that for certain specialisations of the variables and parameters turn out to be Hermite–Padé approximants either of the polylogarithms or of Hurwitz zeta functions. In the former case, we recover known results, while in the latter the results are new and generalise some recent works of Beukers and Prévost. Finally, we make a detailed comparison of our work with Beukers’. Such constructions are useful in the arithmetical study of the values of the Riemann zeta function at integer points and of the Kubota–Leopold p$p$ -adic zeta function.