Note on the best approximation in $L^1$ metric

TL;DR

This paper explores methods for best approximation of functions by trigonometric polynomials in the L^1 metric and applies these to determine optimal constants in inequalities related to Lipschitz function approximation.

Contribution

It introduces an approach for best approximation in L^1 and applies it to find optimal constants in Nikolsky's type inequalities for Lipschitz functions.

Findings

Derived optimal constants in Nikolsky's inequalities

Proposed a new approach for L^1 approximation

Applied method to Lipschitz function approximation

Abstract

We present one of the approaches to find the best approximation of the given function by trigonometric polynomials in $L^1$ metric and applied it to find the optimal constants in the Nikolsky's type inequality, concerning approximation of Lipschitz functions by algebraic polynomials.