This article is cited in 4 papers

Approximation of Classes $B^r_{p,\theta}$ of Periodic Functions of One and Several Variables

A. S. Romanyuk

Institute of Mathematics, Ukrainian National Academy of Sciences

Abstract: We obtain order-sharp estimates of best approximations to the classes $B^r_{p,\theta}$ of periodic functions of several variables in the space $L_q$, $1\le p,q\le\infty$, by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes $B^{r_1}_{1,\theta}$ in the space $L_1$.

Keywords: class $B^r_{p,\theta}$ of periodic functions, trigonometric polynomial, hyperbolic cross, Bernoulli kernel, Fourier hyperbolic sum, Valée-Poussin kernel, Fejér kernel.

UDC: 517.51

Received: 29.01.2008

DOI: 10.4213/mzm4508

 English version: Mathematical Notes, 2010, 87:3, 403–415

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