Related papers: Whitney's theorem for local anisotropic polynomial…

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in $L_p(Q)$ with $1\leq p\leq \infty$. Here $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the coordinate axes. We consider the…

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one…

We survey the main results of approximation theory for adaptive piecewise polynomial functions. In such methods, the partition on which the piecewise polynomial approximation is defined is not fixed in advance, but adapted to the given…

We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly…

We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spacial moduli of smoothness and their properties. In particular, we…

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is…

For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq…

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L_2$ on a compact space, Gr\"ochenig in \cite{G} discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all $1\le p\le\infty$, we…

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w$ becomes infinite) on an interval and not too ``rapidly…

Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on…

It is a classical result in rational approximation theory that certain non-smooth or singular functions, such as $|x|$ and $x^{1/p}$, can be efficiently approximated using rational functions with root-exponential convergence in terms of…

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is…

Among other things, we prove that, for a doubling weight $w$, $0< p\leq\infty$, $r\in{\mathbb N}_0$, and $0<\alpha <r+1 - 1/\lambda_p$, we have \[ E_n(f)_{p, w_n} = O(n^{-\alpha}) \iff \omega_\varphi^{r+1}(f, n^{-1})_{p, w_n} =…

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" refers to (finitely many changes of) monotonicity, convexity,…

An important quantity associated with a complex polynomial $p(z)$ is $\Vert p \Vert_\infty$, the maximum of its modulus over the unit disc $D$. We prove, $z_* \in D$ is a local maximum of $|p(z)|$ if and only if $a_*$ satisfies,…

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, $H^p$ ($1 < p < \infty$). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if…

We study the approximation of $L_p$-functions, $p\in (0,\infty]$, on cylindrical space-time domains $\Omega_T:=[0,T]\times \Omega$, $0<T<\infty$, $\Omega\subset \R^d$ Lipschitz, $d\in \mathbb{N}$, with respect to continuous anisotropic…

We show that the rate of convergence on the approximation of volumes of a convex symmetric polytope P in R^n by its dual L_{p$-centroid bodies is independent of the geometry of P. In particular we show that if P has volume 1,…

We obtain the asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with the equidistant nodes $x_k^{(n-1)}=\frac{2k\pi}{2n-1},\ k\in\mathbb{Z},$ in metrics of the spaces $L_p$ on…

In this work we prove optimal $W^{s,p}$-approximation estimates (with $p\in[1,+\infty]$) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont--Scott approximation theory together with two novel…