# Approximation by translates of a single function of functions in space induced by the convolution with a given function

## TL;DR

This paper investigates how well functions can be approximated using linear combinations of translates of a single function, providing bounds on approximation rates in various $L_p$ spaces for periodic functions induced by convolution.

## Contribution

It introduces new methods for approximation and establishes upper and lower bounds for the approximation rates in $L_p$ spaces, advancing understanding of convolution-induced function classes.

## Key findings

- Established upper bounds for $L_p$ approximation convergence rates.
- Derived lower bounds for the best approximation in the case $p=2$.
- Proposed methods for approximation of convolution-induced function classes.

## Abstract

We study approximation by arbitrary linear combinations of $n$ translates of a single function of periodic functions. We construct some methods of this approximation for functions in a class induced by the convolution with a given function, and prove upper bounds of $L_p$-the approximation convergence rate by these methods, when $n \to \infty$, for $1 < p < \infty$, and lower bounds of the quantity of best approximation of this class by arbitrary linear combinations of $n$ translates of arbitrary function, for the particular case $p=2$.