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

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