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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 .…”

“…functions. In the spaces S p of functions of one and several variables, Theorem 1 and Corollaries 1, 3 and 4 were proved in [28] and [2], respectively. In the spaces L 2 , for classical moduli of smoothness inequality (19) was proved by N.I.…”

“…Proof. It was shown in [28] that I n (s) ≥ 2 when s ≥ 1 and I n (s) ≥ 1 + 2 s−1 when s ∈ (0, 1). Combining these two estimates and ( 26), we get (28).…”

“…It was shown in [28] that I n (s) ≥ 2 when s ≥ 1 and I n (s) ≥ 1 + 2 s−1 when s ∈ (0, 1). Combining these two estimates and ( 26), we get (28). Relation (29) follows from the estimate I n (mp/2) ≥ 1 + 1/ √ 2, which is a consequence of the above estimates for the value of I n (s) in the case when m ∈ N and 1 ≤ p < ∞ (see [28]).…”

“…11], etc. In [28], direct and inverse theorems for the approximation of functions from the spaces S p were proved in terms of their best approximations by trigonometric polynomials and moduli of smoothness of arbitrary positive orders. In [3], exact Jackson-type inequalities in the spaces S p were obtained in terms of the best approximations of functions and the averaged values of their generalized moduli of smoothness as well as the exact values were found for widths of classes of 2π-periodic functions defined by certain conditions on the averaged values of their generalized moduli of smoothness.…”