After the appearance of the article “A retrospective on 60 years of approximation theory and associated fields” (J. Approx. Theory 160 (1–2) (2009) 3–18), several readers informed me (PLB) that they would like to see an article based upon research contacts and conference participations of members of the chair “Lehrstuhl A für Mathematik” at Aachen throughout the world. The present paper is devoted

This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real

We correct a mistake in the statement of Theorem 2.4.

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because

We show that if U is an arbitrary open subset of a Riemann surface and φ an arbitrary continuous function on the boundary ∂U, then there exists a holomorphic function φ˜ on U such that, for every p∈∂U, φ˜(x)→φ(p), as x→p outside a set of density 0 at p relative to U. These “solutions to a boundary problem” are not unique. In fact they can be required to have interpolating properties and also to assume

In this paper we prove that if a multivariate function of a certain smoothness class is represented by a sum of k arbitrarily behaved ridge functions, then it can be represented by a sum of k ridge functions of the same smoothness class and a polynomial of degree at most k−1. This solves the problem posed by A. Pinkus in his monograph “Ridge Functions” up to a multivariate polynomial.

It is known that for a ϱ-weighted Lq approximation of single variable functions defined on a finite or infinite interval, whose rth derivatives are in a ψ-weighted Lp space, the minimal error of approximations that use n samples of f is proportional to ‖ω1∕α‖L1α‖f(r)ψ‖Lpn−r+(1∕p−1∕q)+, where ω=ϱ∕ψ and α=r−1∕p+1∕q, provided that ‖ω1∕α‖L1<+∞. Moreover, the optimal sample points are determined by quantiles

Best approximation by the set of all n-fold linear combinations of a family of characteristic functions of measurable subsets is investigated. Such combinations generalize Heaviside-type neural networks. Existence of best approximation is studied in terms of approximative compactness, which requires convergence of distance-minimizing sequences. We show that for (Ω,μ) a measure space, in Lp(Ω,μ) with

We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of the appropriate Blaschke product at points on the unit circle.

Identifying (conditionally) strictly positive definite functions is of great importance as they allow the unique solution of certain interpolation problems. We introduce new sufficient (and some necessary) conditions for functions to be conditionally strictly positive definite on all spheres Sd−1, d>2, only employing monotonicity properties. For strictly positive definite and conditionally negative

In this article, we focus on the connections of monotonicity properties and the strict positive definiteness of functions on Rd. We collect the existing results known for the Euclidean space and present a new technique to construct positive definite functions from multiply monotone functions. Further, we collect properties of multiply monotone functions which allow us to construct new positive definite

The holomorphic extensions Fα(iz), z∈ℂd, of the Fourier transforms Fα≔f̂α of a sequence of continuous real functions fα(x), x∈Rd,α∈N0d, with exponential decay at infinity, generates a sequence of polynomials Qβ, β∈N0d, of degree |β|, that are biorthogonal to the distributional derivatives μα≔(−1)|α|fα(α), α∈N0d. The generating function is a generalized Taylor series expansion of the Fourier-Laplace

In this paper, we discuss various basic properties of moduli of smoothness of functions from Lp(Rd), 0

Four Jacobi settings are considered in the context of Hardy’s inequality: the trigonometric polynomials and functions, and the corresponding symmetrized systems. In the polynomial cases sharp Hardy’s inequality is proved for the type parameters α,β∈(−1,∞)d, whereas in the function systems for α,β∈[−1∕2,∞)d. The ranges of these parameters are the widest in which the corresponding orthonormal bases are

We study the entropy numbers of the compact embeddings between smoothness Morrey spaces on bounded domains. Here we discover a new phenomenon when the difference of smoothness parameters in the source and target spaces is rather small compared with the influence of the fine parameters in the Morrey setting. In view of some partial forerunners this was not to be expected till now. Our argument relies

Let E be a closed subset of the unit circle of measure zero. Recently, Beise and Müller showed the existence of a function in the Hardy space H2 for which the partial sums of its Taylor series approximate any continuous function on E. In this paper, we establish an analogue of this result in a non-linear setting where we consider optimal polynomial approximants of reciprocals of functions in H2 instead

For weighted L2 spaces with doubling weights w on [−1,1] and norms ‖⋅‖2,w, the Markov factor on a polynomial P is defined by ‖P′‖2,w‖P‖2,w. We study this Markov factor on random polynomials with independent N(0,1) coefficients, and show that the upper bound of the average (expected) Markov factor is order degree to the 3∕2, as compared to the degree squared worst case upper bound.

We study bracketing covering numbers for spaces of bounded convex functions in the Lp norms. Bracketing numbers are crucial quantities for understanding asymptotic behavior for many statistical nonparametric estimators. Bracketing number upper bounds in the supremum distance are known for bounded classes that also have a fixed Lipschitz constraint. However, in most settings of interest, the classes

A limiting property of the nearest-neighbor recurrence coefficients for multiple orthogonal polynomials from a Nevai class is investigated. Namely, assuming that the nearest-neighbor coefficients have a limit along rays of the lattice, we describe it in terms of the solution of a system of partial differential equations. In the case of two orthogonality measures the differential equation becomes ordinary

We study a family of polynomials which are orthogonal with respect to the varying, highly oscillatory complex weight function eniλz on [−1,1], where λ is a positive parameter. This family of polynomials has appeared in the literature recently in connection with complex quadrature rules, and their asymptotics have been previously studied when λ is smaller than a certain critical value, λc. Our main

We study the volume of unit balls Bp,qn of finite-dimensional Lorentz sequence spaces ℓp,qn. We give an iterative formula for vol(Bp,qn) for the weak Lebesgue spaces with q=∞ and explicit formulas for q=1 and q=∞. We derive asymptotic results for the nth root of vol(Bp,qn) and show that [vol(Bp,qn)]1∕n≍p,qn−1∕p for all 0

Semialgebraic splines are bivariate splines over meshes whose edges are arcs of algebraic curves. They were first considered by Wang, Chui, and Stiller. We compute the dimension of the space of semialgebraic splines in two extreme cases. If the polynomials defining the edges span a three-dimensional space of polynomials, then we compute the dimensions from the dimensions for a corresponding rectilinear

In this paper we solve Padé- (i.e. multiple) and Prony (i.e. simple exponential) interpolation problems for the generalized exponential sums with equal weights: Hn(z;h)≔μn∑k=1nh(λkz),whereμ,λk∈ℂ,and h is a fixed analytic function under few natural assumptions. The interpolation of a function f by Hn is due to properly chosen μ and {λk}k=1n, which depend on f, h and n. The sums Hn are related to the

We study the minimum of the essential spectrum of canonical systems Ju′=−zHu. Our results can be described as a generalized and more quantitative version of the characterization of systems with purely discrete spectrum, which was recently obtained by Romanov and Woracek (2020). Our key tool is oscillation theory.

We consider orthogonal polynomials pne−2nQn,x for varying measures and use universality limits to prove ”local limits” limn→∞pne−2nQn,yjn+zK̃nyjn,yjnpne−2nQn,yjne−nQn′yjnK̃nyjn,yjnz=cosπz.Here yjn is a local maximum point of pne−nQn in the ”bulk” of the support, K̃nyjn,yjn is the normalized reproducing kernel, and the limit holds uniformly for z in compact subsets of the plane. We also consider local

The aim of the paper is to establish (local) optimal embeddings of Besov spaces Bp,r0,b involving only a slowly varying smoothness b. In general, our target spaces are outside of the scale of Lorentz–Karamata spaces. In particular, we improve results from Caetano (2011), where the targets are (local) Lorentz–Karamata spaces. To derive such results, we apply limiting real interpolation techniques and

We study the (periodic) autocorrelation coefficients of the Rudin-Shapiro polynomials and prove that Theorem. If Pn and Qn are the nth Rudin-Shapiro polynomials and |Pn(z)|2=∑k=−Ln+1Ln−1akzk,|z|=1 (a0=Ln, ak=a−k, k≥1), then 0.27771487⋯(1+o(1))|λ|n≤max1≤k≤Ln−1|ak|≤(3.78207844⋯)|λ|nwhere λ is the real root of x3−x2−2x+4=0. Also, if we let Pn¯Qn(z)=PnQn¯(1∕z)=∑k=−Ln+1Ln−1bkzk,|z|=1 (b0=2−Ln, bk=b−k¯,

We are interested in functions analytic in the unit disc D of the complex plane with a wild behaviour near the boundary T of D. For instance, the main result implies the existence of a residual subset of H(D) every element f of which satisfies the property that, given any compact subset K of T, different from T, any continuous function φ on K, and any compact set L of D, there exists an increasing

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by rGν(λ)(x) (right GGF-Fs) and lGν(λ)(x) (left GGF-Fs) with x∈(−1,1), λ>−1∕2 and real ν≥0, are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable

Given a bivariate function and a finite rectangular grid, we perform transfinite interpolation at all the points on the grid lines. By noting the uniqueness of interpolation by rank-n functions, we prove that the result is identical to the output of Schneider’s CA2D algorithm (schneider, 2010) . Furthermore, we use the tensor-product version of bivariate divided differences to derive a new error bound

We consider polynomials of degree d with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of such polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of dth powers of two

We obtain the following dimension independent Bernstein–Markov inequality in Gauss space: for each 1≤p<∞ there exists a constant Cp>0 such that for any k≥1 and all polynomials P on Rk we have ‖∇P‖Lp(Rk,dγk)≤Cp(degP)12+1πarctan|p−2|2p−1‖P‖Lp(Rk,dγk),where dγk is the standard Gaussian measure on Rk. We also show that under some mild growth assumptions on any function B∈C2((0,∞))∩C([0,∞)) with B′,B′′>0

This paper is about the characterization of the rate of best polynomial approximation on a domain in Rd by multivariate polynomials of degree n via a suitable modulus of continuity or smoothness. We shall be concerned with the situations: approximation on convex polytopes and approximation on (not necessarily convex) smooth algebraic domains. We shall introduce appropriate moduli of smoothness and

We prove sharp inequalities between Lp−norms (1

Consider the Hermite operator H=−Δ+|x|2 on the Euclidean space Rn. The main aim of this article is to prove the boundedness of Hermite pseudo-multiplier σ(⋅,H) on some appropriate Besov and Triebel–Lizorkin spaces.

Let V⊂Rm be a centrally symmetric convex body and let V∗⊂Rm be its polar. We prove limit relations between the sharp constants in the multivariate Markov–Bernstein–Nikolskii type inequalities for algebraic polynomials on V∗ and the corresponding constants for entire functions of exponential type with the spectrum in V.

We derive a formula for weighted polynomial least squares approximation which expresses the approximant as a convex combination of interpolants. There is a similar formula for L2 approximation and the same principle applies to multivariate approximation.

We construct an analytic self-map Φ of the bidisk D2 whose image touches the distinguished boundary, but whose approximation numbers of the associated composition operator on H2(D2) are small in the sense that lim supn→∞[an2(CΦ)]1∕n<1.

Given a closed set E of Lebesgue measure zero on the unit circle T there is a continuous function f on T such that for every continuous function g on E there is a subsequence of partial Fourier sums Sn+(f,ζ)=∑k=0nfˆ(k)ζkof f, which converges to g uniformly on E. This result completes one result in a recent paper by C. Papachristodoulos and M. Papadimitrakis (2019), see (Papachristodoulos and Papadimitrakis

We complete a gap in the proof that exceptional polynomials are complete orthogonal systems in the associated Hilbert spaces.

Exceptional orthogonal polynomials are complete families of orthogonal polynomials that arise as eigenfunctions of a Sturm–Liouville problem. Antonio Durán discovered a gap in the original proof of completeness for exceptional Hermite polynomials, that has propagated to analogous results for other exceptional families. In this paper we provide an alternative proof that follows essentially the same

This paper deals with a study of En(f)Lw2(S), the rate of approximation by polynomials of total degree n in terms of the norm of Lw2(S) where S is the simplex S={(x1,…,xd):xi≥0,x0=1−∑i=1dxi≥0}andw=wγ=x0γ0…xdγd,γi>−1. Direct and converse results are achieved relating En(f)Lw2(S) to the operators φξr(x)(∂∂ξ)rf and corresponding K-functionals, where ξ∈E(S) and E(S) are the edges of S. For the special

We correct a mistake in the statements of Corollaries 4.11 and 4.12. The mistake has no bearing to their proofs or other results in the paper.

We prove the existence of series [Formula: see text], whose coefficients [Formula: see text] are in [Formula: see text] and whose terms [Formula: see text] are translates by rational vectors in [Formula: see text] of a family of approximations to the identity, having the property that the partial sums are dense in various spaces of functions such as Wiener's algebra [Formula: see text], [Formula: see