Abstract
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.
Highlights
The fundamental problem in approximation theory is to find for a complicated function f in a quasinormed space X a close-by, simple approximant Pn from a subset of X such that the error of approximation f − Pn X can be controlled by a specific majorant
We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes
We illustrate this by considering the well-known case of approximation of periodic functions by trigonometric polynomials on T = [0, 2π]
Summary
The fundamental problem in approximation theory is to find for a complicated function f in a quasinormed space X a close-by, simple approximant Pn from a subset of X such that the error of approximation f − Pn X can be controlled by a specific majorant. If f ∈ Lp(T), 1 ≤ p ≤ ∞, and 0 < α < r, for the best approximant Tn∗ and the modulus of smoothness ωr(f, t)p, the following conditions are equivalent:. Let us emphasize that this result holds under very mild conditions on the approximants P2n (f ) These inequalities imply the results similar to those given in the equivalence (i2) ⇔ (i3). We study smoothness properties of algebraic polynomials and splines of the best approximation and consider some Fourier means related to Fourier–Jacobi series. If F G and G F simultaneously, we write F G and say that F is equivalent to G
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