# Inequalities of Jackson-Stechkin type for elements of Hilbert space (in Russian)

## TL;DR

This paper introduces a new generalized modulus of continuity for Hilbert space elements and derives exact Jackson-Stechkin type inequalities, unifying and extending classical approximation results for various function classes.

## Contribution

It proposes a novel generalized modulus of continuity and establishes new exact inequalities of Jackson-Stechkin type for Hilbert space elements, encompassing many classical approximation inequalities.

## Key findings

- New generalized modulus of continuity for Hilbert space elements
- Exact Jackson-Stechkin type inequalities derived using the new modulus
- Results include and extend classical approximation inequalities

## Abstract

In this paper we introduced a new characteristics of the elements of a Hilbert space - generalized moduli of continuity $\omega_\varphi(x;L_{p,V}([0,\delta]))$ and obtain new exact inequalities of Jackson - Stechkin type with these moduli of continuity for the approximation of elements of a Hilbert space. These results include many well-known inequalities for approximation of periodic functions by trigonometric polynomials, approximation of non-periodic functions by entire functions of exponential type, similar results for almost periodic functions, and others. A number of results is new even in these classic situations.