2019
Том 71
№ 11

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Approximations in spaces of locally integrable functions

Stepanets O. I.

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Abstract

We study approximations of functions from the sets $\hat{L}^{\psi}_{\beta}\mathfrak{N}$, which are determined by convolutions of the following form: $$f(x) = A_0 + \int\limits_{-\infty}^{+\infty}\varphi(x + t) \hat{\psi}_{\beta}(f)dt, \quad \varphi \in \mathfrak{N},\quad \hat{\psi}_{\beta} \in L(-\infty, +\infty)$$ where $\mathfrak{N}$ is a fixed subset of functions with locally integrable $p$-th powers $(p \geq 1)$. As an approximating aggregate, we use so-called Fourier operators, which are entire functions of the exponential type $\leq \sigma$ that turn into trigonometric polynomials if the function $\varphi(\cdot)$ is periodic (in particular, they may be the Fourier sums of the function approximated). Approximations are studied in the spaces $\hat{L}_p$ determined by a locally integrable norm $||\cdot||_{\hat{p}}$. Analogs of the Lebesgue and Favard inequalities, well-known in the periodic case, are obtained and used for finding order-exact estimates of the corresponding best approximations and estimates of approximations by Fourier operators, which are order-exact and, in some important cases, they arc also exact in the sense of constants with principal terms of these estimates.

English version (Springer): Ukrainian Mathematical Journal 46 (1994), no. 5, pp 638-670.

Citation Example: Stepanets O. I. Approximations in spaces of locally integrable functions // Ukr. Mat. Zh. - 1994. - 46, № 5. - pp. 597–625.

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