Smoothness of functions vs. smoothness of approximation processes

2019/03/01 by Yurii Kolomoitsev, Sergey Tikhonov, Kolomoitsev, Yu. +1 · 1 citation
Mathematics · #41A15 #41A17 #41A25 (Secondary) #41A35 #41A50 #41A63 #41A65 #42A45 #42B25 (Primary) #Advanced Harmonic Analysis Research #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1903.00229

openalex publication_date 2019/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 K-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on \mathbbT^d, ℝ^d, [-1, 1], nonlinear wavelet approximation, etc.

Cited by

• Special measures of smoothness for approximation by sampling operators in L_p(ℝ^d)

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