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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

Sharp constants for approximations of convolution classes with an integrable kernel by spaces of shifts


Author: O. L. Vinogradov
Translated by: the author
Original publication: Algebra i Analiz, tom 30 (2018), nomer 5.
Journal: St. Petersburg Math. J. 30 (2019), 841-867
MSC (2010): Primary 41A17; Secondary 41A44
DOI: https://doi.org/10.1090/spmj/1572
Published electronically: July 26, 2019
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Abstract: Let $ \sigma >0$, and let $ G,B\in L(\mathbb{R})$. This paper is devoted to approximation of convolution classes $ f=\varphi *G$, $ \varphi \in L_p(\mathbb{R})$, by a space  $ \mathbf S_B$ that 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$ and $ B$, linear operators $ {\mathcal X}_{ \sigma ,G ,B}$ with values in  $ \mathbf S_B$ are constructed for which

$\displaystyle \Vert f-{\mathcal X}_{\sigma ,G,B}(f)\Vert _p\leq {\mathcal K}_{\sigma ,G}\Vert\varphi \Vert _p.$    

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.

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Additional Information

O. L. Vinogradov
Affiliation: St. Petersburg State University, Universitetskii pr. 28, 198504 St. Petersburg, Russia
Email: olvin@math.spbu.ru

DOI: https://doi.org/10.1090/spmj/1572
Keywords: Spaces of shifts, sharp constants, convolution, Akhiezer--Kre\u{\i}n--Favard type inequalities
Received by editor(s): June 30, 2018
Published electronically: July 26, 2019
Additional Notes: This work is supported by the Russian Science Foundation under grant no. 18-11-00055
Article copyright: © Copyright 2019 American Mathematical Society