Approximation by linear combinations of translates of a single function

TL;DR

This paper investigates the approximation of periodic functions using linear combinations of translates of a single function, providing convergence rates and bounds for univariate and multivariate cases.

Contribution

It introduces new linear approximation methods for periodic functions and establishes upper and lower bounds for approximation errors in various function classes.

Findings

Established $L^p$-approximation convergence rates as $n o fty$

Constructed linear methods for univariate and multivariate functions

Proved bounds for best approximation using translates of a single function

Abstract

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the $L^p$-approximation convergence rate by these methods, when $n \to \infty$, for $1 \leq p \leq \infty$. We also generalize these results to classes of multivariate functions defined the convolution with the tensor product of a single function. In the case $p=2$, for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of $n$ translates of arbitrary function.