Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces

Abstract

In Musilak-Orlicz type spaces S M , direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

“…In the spaces L 2 of 2π-periodic square-summable functions, for moduli of continuity, this result was obtained by Babenko [4]. In the spaces S p of functions of one and several variables, this result for classical moduli of smoothness was obtained, respectively, [28] and [2], and for generalized moduli of smoothness, in [1] (for functions of one variable). In the proof of Theorem 1, we mainly use the ideas outlined in [28,4,14,15], taking into account the peculiarities of the spaces BS p .…”

“…Proof. Let us use the scheme of the proof from [28,1], taking into account the peculiarities of the spaces BS p . As above, for any f ∈ BS p , ϕ ∈ Φ and h ∈ R, we denote by ∆ ϕ h f p the usual norm (2) of the function ∆ ϕ h f satisfying relation (10) (if such a function ∆ ϕ h f ∈ B-a.p.…”

“…In spaces of almost periodic functions, direct approximation theorems were established in the papers [11,23,24,6], etc. In particular, in [23], an analogue of the inequality (1) was obtained for Besicovitch almost periodic functions of the order 2 (B 2 -a.p. functions).…”

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

“…In the spaces L 2 of 2π-periodic square-summable functions, for moduli of continuity, this result was obtained by A.G. Babenko [4]. In the spaces S p of functions of one and several variables, this result for classical moduli of smoothness was obtained in [28] and [2], respectively, and for generalized moduli of smoothness, in [1] (for functions of one variable). In the proof of Theorem 1, we mainly use the ideas outlined in [4,14,15,28], taking into account the peculiarities of the spaces BS p .…”

“…Proof. Let us use the scheme of the proof from [1,28], taking into account the peculiarities of the spaces BS p . As above, for any f ∈ BS p , ϕ ∈ Φ and h ∈ R, we denote by ∆ ϕ h f p the usual norm (2) of the function ∆ ϕ h f satisfying relation ( 9) (if such a function ∆ ϕ h f ∈ B-a.p.…”