$L$-approximation of $B$-splines by trigonometric polynomials

TL;DR

This paper investigates the approximation of B-splines by trigonometric polynomials in the L-norm, extending previous work on characteristic functions, and provides new results including sharpness proofs for specific cases.

Contribution

It introduces new results on L-approximation of B-splines and proves sharpness for particular parameter values, advancing understanding in approximation theory.

Findings

Derived simple results on L-approximation of B-splines.

Proved the sharpness of approximation bounds for specific h values.

Extended previous work on characteristic functions to B-splines.

Abstract

This note is a continuation of our papers [1,2], devoted to $L$-approximation of characteristic function of $(-h, h)$ by trigonometric polynomials. In the paper [1] the sharp values of the best approximation for the special values of $h$ were found. In [2] we gave the complete solution of the problem for arbitrary values of $h$. In general case [2] the situation is more deep and results are not so simple as in [1]. For applications to the problem of optimal constants in the Jackson-type inequalities we need, however, results on $L$-approximation of $B$-splines and linear combinations of $B$-splines. Here we present some simple results about $L$-approximation of $B$-splines as well as give the the proof of its sharpness for the special values of $h$.