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 prove the corresponding Jackson theorem. The usual weak converses are also true.

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, 2019). They proved that for a classical one third Cantor set C there is no universal function in the disk algebra. They also proved that for a symmetric Cantor set C∗ on T there is no universal continuous function for the classical symmetric Fourier sums.

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 arguments, but provides a direct proof of the key lemma on which the completeness proof is based. This direct proof makes use of the theory of trivial monodromy potentials developed by Duistermaat and Grünbaum and Oblomkov.

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 case that S is the triangle somewhat weaker results were recently achieved in Feng et al. (2019).

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 text], [Formula: see text], [Formula: see text], for every [Formula: see text], and the space of measurable functions. Applying this theory to particular situations, we establish approximations by such series to solutions of the heat and Laplace equations as well as to probability density functions.