Sharp estimates for mean square approximations of classes of periodic convolutions by spaces of shifts,St. Petersburg Mathematical Journal

当前位置： X-MOL 学术 › St. Petersburg Math. J. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

Sharp estimates for mean square approximations of classes of periodic convolutions by spaces of shifts
St. Petersburg Mathematical Journal ( IF 0.8 ) Pub Date : 2021-03-02 , DOI: 10.1090/spmj/1650
A. Yu. Ulitskaya

Abstract:Let

be the classical Lebesgue spaces of $2\pi$ -periodic functions and

the best approximation of

by the space

. For $n\in \mathbb{N}$ , $B\in L_2$ , the symbol $\mathbb{S}_{B,n}$ stands for the space of functions

of the form

$\displaystyle s(x)=\sum _{j=0}^{2n-1}\beta _jB\Big (x-\frac {j\pi }{n}\Big ).$

In this paper, all spaces $\mathbb{S}_{B,n}$ are described that provide a sharp constant in several inequalities for approximation of classes of convolutions with a kernel $G\in L_1$ . In particular, necessary and sufficient conditions are obtained under which the inequality

$\displaystyle E\bigl (f,\mathbb{S}_{B,n}\bigr )_2\leq \vert c^\ast _{2n+1}(G)\vert\Vert\varphi \Vert _2$

is fulfilled. This inequality is sharp on the class of functions

representable in the form $f=G\ast \varphi$ , $\varphi \in L_2$ . The constant $\vert c^\ast _{2n+1}(G)\vert$ is the

th term of the sequence $\{\vert c_l(G)\vert\}_{l\in \mathbb{Z}}$ of absolute values of the Fourier coefficients of

arranged in nonincreasing order. In addition, easily verifiable conditions are indicated that suffice for the above inequality. Examples of kernels and extremal subspaces satisfying these conditions are provided.

中文翻译：

对周期卷积类别的均方逼近，移位空间的敏锐估计

摘要：设

是经典的勒贝格空间 $2 \ pi$ 为周期函数和

的最佳逼近

由空间

在

。对于，符号代表形式的功能空间 $n \ in \ mathbb {N}$ $B \ in L_2$ $\ mathbb {S} _ {B，n}$

$\ displaystyle s（x）= \ sum _ {j = 0} ^ {2n-1} \ beta _jB \ Big（x- \ frac {j \ pi} {n} \ Big）。$

在本文中，描述了所有空间，这些空间在几个不等式中提供了一个尖锐的常数，以近似于带核的卷积类。特别是，获得了不平等的必要条件和充分条件 $\ mathbb {S} _ {B，n}$ $G \ in L_1$

$\ displaystyle E \ bigl（f，\ mathbb {S} _ {B，n} \ bigr）_2 \ leq \ vert c ^ \ ast _ {2n + 1}（G）\ vert \ Vert \ varphi \ Vert _2$

完成。这个不等式是在类的功能急剧

在形式表示，。常数是按非递增顺序排列的傅立叶系数的绝对值序列的第th个项。另外，表明容易验证的条件足以满足上述不等式。提供了满足这些条件的核和极值子空间的示例。 $f = G \ ast \ varphi$ $\ varphi \ in L_2$ $\ vert c ^ \ ast _ {2n + 1}（G）\ vert$

$\ {\ vert c_l（G）\ vert \} _ {l \ in \ mathbb {Z}}$

更新日期：2021-03-04

点击分享查看原文

点击收藏

阅读更多本刊最新论文

全部期刊列表>>