On the Conditions for the Embedding of Classes of Besicovitch Almost Periodic Functions

Khasanov, Yu. Kh.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 1.

Abstract:
In the paper we  established  some conditions for embedding of  classes of \(B_q\)-almost-periodic functions into the classes of  \(B_p\)-almost-periodic in the sense of Besicovitch functions with
arbitrary Fourier exponents for \({1\leq p<q<\infty}\). Some of established conditions are counterparts of the known results of  other authors on embedding of the classes \(L_p\) \((1\leq p<\infty)\)
of periodic functions. As a structural characteristic of such functions we use a higher-order modulus of smoothness with a predetermined step. Since the space of almost periodic Besicovitch
functions is a complete normed space, the Bochner-Fejer polynomials are used as polynomials of best approximation. We also indicate some conditions for the Besicovitch functions to belong to the class of entire functions of bounded degree. It is established that if a \(B_p\)-almost periodic \(f(x)\in B_p\) has the best approximation value by entire functions of bounded degree, then there exists the
absolutely continuous derivative of the function which is also \(B_p\)-almost periodic.

Keywords: Besicovitch almost periodic functions, Fourier series, trigonometric polynomials, embedding theorems, spectral function, modulus of continuity, entire function, Bochner-Fejer polynomials

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