On the optimal constants in the two-sided Stechkin inequalities

TL;DR

This paper investigates the best possible constants in the strong and weak Stechkin inequalities, providing improved bounds and elementary proofs, with implications for approximation spaces and interpolation theory.

Contribution

It offers new optimal constants for both discrete and continuous Stechkin inequalities and simplifies the proof for a key constant in the strong discrete case.

Findings

Improved constants for the strong discrete Stechkin inequality

Elementary proof of Bennett's constant in the strong discrete case

Minimal constants for the weak discrete and continuous Stechkin inequalities

Abstract

We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson. Finally, the minimal constants in the weak discrete Stechkin inequalities and both continuous Stechkin inequalities are presented.