Classes of convolutions with a singular family of kernels: Sharp constants for approximation by spaces of shifts

Vinogradov, O.

doi:10.1090/spmj/1646

Classes of convolutions with a singular family of kernels: Sharp constants for approximation by spaces of shifts
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by O. L. Vinogradov
Translated by: O. L. Vinogradov

St. Petersburg Math. J. 32 (2021), 233-260

DOI: https://doi.org/10.1090/spmj/1646

Published electronically: March 2, 2021

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Abstract:

Let $\sigma >0$, and let $G,B\in L(\mathbb R)$. The paper is devoted to approximation of classes of functions $f$ for every $\varepsilon >0$ representable as \begin{equation*} f(x)=F_{\varepsilon }(x)+ \frac {1}{2\pi }\int _{\mathbb R}\varphi (t)G_{\varepsilon }(x-t) dt, \end{equation*} where $F_{\varepsilon }$ is an entire function of type not exceeding $\varepsilon$, $G_{\varepsilon }\in L(\mathbb R)$, and $\varphi \in L_p(\mathbb R)$. The approximating space $\mathbf S_B$ consists of functions of the form \begin{equation*} s(x)=\sum _{j\in \mathbb Z}\beta _jB\Big (x-\frac {j\pi }{\sigma }\Big ). \end{equation*} Under some conditions on $G=\{G_{\varepsilon }\}$ and $B$, linear operators ${\mathcal X}_{\sigma ,G,B}$ with values in $\mathbf S_B$ are constructed for which $\|f-{\mathcal X}_{\sigma ,G,B}(f)\|_p\leq {\mathcal K}_{\sigma ,G}\|\varphi \|_p$. For $p=1,\infty$ the constant ${\mathcal K}_{\sigma ,G}$ (it is an analog of the well-known Favard constant) cannot be reduced, even if one replaces the left-hand side by the best approximation by the space $\mathbf S_B$. The results of the paper generalize classical inequalities for approximations by entire functions of exponential type and by splines.

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