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Approximation via Hausdorff operators

Part of: Integral transforms, operational calculus Harmonic analysis in one variable Approximations and expansions

Published online by Cambridge University Press:  13 August 2020

Alberto Debernardi  and
Elijah Liflyand
Alberto Debernardi
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel, 52900 e-mail: adebernardipinos@gmail.com
Elijah Liflyand*
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel, 52900 and Regional Mathematical Center of Southern Federal University, Bolshaya Sadovaya Str. 105/42, Rostov-on-Don, Russia, 344006
*
e-mail: liflyand@math.biu.ac.il
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Abstract

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Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate functions and the adjoint to that Hausdorff operator of the given function. We find estimates for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation of functions and comparison with approximate identities are given in the case of continuous functions from the class $\text {Lip }\alpha $ .

Keywords

Hausdorff operatorsapproximation in Lebesgue spacesmoduli of continuity

MSC classification

Primary: 41A25: Rate of convergence, degree of approximation
Secondary: 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 44A15: Special transforms (Legendre, Hilbert, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 512 - 529
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Canadian Mathematical Society 2020

Footnotes

Alberto Debernardi was supported by the ERC starting grant No. 713927 and the ISF grant No. 447/16. Elijah Liflyand is the corresponding author.

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