Some exact constants for bilateral approximations in quasi-normed groups

TL;DR

This paper derives precise constants for bilateral approximation inequalities in quasi-normed Abelian groups, providing new theoretical bounds and practical error estimates for approximation methods.

Contribution

It introduces exact constants in bilateral Bernstein-Jackson inequalities for quasi-normed groups, enhancing approximation theory with integral representations and numerical applications.

Findings

Exact constants for approximation inequalities established

Error estimates for spectral and numerical approximations derived

Integral representations for quasi-norms developed

Abstract

We establish inverse and direct theorems on best approximations in quasi-normed Abelian groups through bilateral Bernstein-Jackson inequalities with exact constants. Using integral representations for quasi-norms of functions $f$ in Lebesgue's spaces by decreasing rearrangements $f^*$ with the help of approximation $E$-functionals, error estimates are found. Examples of numerical calculations and spectral approximations of self-adjoint operators obtained by obtained estimates are given.