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

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

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