Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General Settings

TL;DR

This paper introduces a unified framework for sharp constants in approximation theory, establishes conditions for their asymptotic relations, and provides diverse examples to illustrate the approach.

Contribution

It offers a new unified definition of sharp constants and derives conditions for their asymptotic relations in general approximation settings.

Findings

Unified definition of sharp constants introduced

Sufficient conditions for asymptotic relations established

Numerous examples illustrating the approach provided

Abstract

In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Second, we find sufficient conditions that imply limit relations between various sharp constants of approximation theory in general settings. Third, a number of examples from various areas of approximation theory illustrates the general approach.