Jackson–Stechkin-Type Inequalities for the Approximation of Elements of Hilbert Spaces

Abstract

We introduce new characteristics for elements of Hilbert spaces, namely, their generalized moduli of continuity ωφ(x, Lp,V ([0, δ])) and obtain new exact Jackson–Stechkin-type inequalities with these moduli of continuity for the approximation of elements of Hilbert spaces. These results include numerous well-known inequalities for the approximation of periodic functions by trigonometric polynomials, approximation of nonperiodic functions by entire functions of exponential type, similar results for almost periodic functions, etc. Some of these results are new even in these classical cases.