Classes of convolutions with a singular family of kernels: Sharp constants for approximation by spaces of shifts,St. Petersburg Mathematical Journal

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Classes of convolutions with a singular family of kernels: Sharp constants for approximation by spaces of shifts
St. Petersburg Mathematical Journal ( IF 0.7 ) Pub Date : 2021-03-02 , DOI: 10.1090/spmj/1646
O. L. Vinogradov

Abstract:Let $\sigma >0$ , and let $G,B\in L(\mathbb{R})$ . The paper is devoted to approximation of classes of functions

for every $\varepsilon >0$ representable as

$\displaystyle f(x)=F_{\varepsilon }(x)+ \frac {1}{2\pi }\int _{\mathbb{R}}\varphi (t)G_{\varepsilon }(x-t)\,dt,$

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

$\displaystyle s(x)=\sum _{j\in \mathbb{Z}}\beta _jB\Big (x-\frac {j\pi }{\sigma }\Big ).$

Under some conditions on $G=\{G_{\varepsilon }\}$ and

, linear operators ${\mathcal X}_{\sigma ,G,B}$ with values in $\mathbf S_B$ are constructed for which $\Vert f-{\mathcal X}_{\sigma ,G,B}(f)\Vert _p\leq {\mathcal K}_{\sigma ,G}\Vert\varphi \Vert _p$ . 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.

中文翻译：

具有奇异核系列的卷积类别：尖锐常数，用于通过移位空间进行近似

摘要：让 $\ sigma> 0$ ，让。本文专门针对每个可表示为 $G，B \ in L（\ mathbb {R}）$

$\ varepsilon> 0$

$\ displaystyle f（x）= F _ {\ varepsilon}（x）+ \ frac {1} {2 \ pi} \ int _ {\ mathbb {R}} \ varphi（t）G _ {\ varepsilon}（xt） \，dt，$

其中是式的整体功能不超过，和。近似空间由以下形式的函数组成 $F _ {\ varepsilon}$ $\ varepsilon$ $G _ {\ varepsilon} \ in L（\ mathbb {R}）$ $\ varphi \ in L_p（\ mathbb {R}）$ $\ mathbf S_B$

$\ displaystyle s（x）= \ sum _ {j \ in \ mathbb {Z}} \ beta _jB \ Big（x- \ frac {j \ pi} {\ sigma} \ Big）。$

下在某些条件下和，线性算子与值被构造为哪些。因为该常数（它是众所周知的Favard常数的类似物）无法减少，即使用空格按最佳近似值替换了左手边。本文的结果通过指数类型的整体函数和样条曲线推广了近似的经典不等式。 $G = \ {G _ {\ varepsilon} \}$

${\数学X} _ {\ sigma，G，B}$ $\ mathbf S_B$ $\ Vert f-{\ mathcal X} _ {\ sigma，G，B}（f）\ Vert _p \ leq {\ mathcal K} _ {\ sigma，G} \ Vert \ varphi \ Vert _p$ $p = 1，\ infty$ ${\数学K} _ {\ sigma，G}$ $\ mathbf S_B$

更新日期：2021-03-04

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