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Ufimsk. Mat. Zh., 2018, Volume 10, Issue 1, Pages 96–117 (Mi ufa421)  

On two-sided estimate for norm of Fourier operator

I. A. Shakirov

Naberezhnye Chelny State Pedagogical University, Nizametdinova str. 28, 423806, Naberezhnye Chelny, Russia

Abstract: In the work we study the behavior of Lebesgue constant $L_n$ of the Fourier operator defined in the space of continuous $2\pi$-periodic functions. The known integral representations expressed in terms of the improper integrals are too cumbersome. They are complicated both for theoretical and practical purposes.
We obtain a new integral representation for $L_n$ as a sum of Riemann integrals defined on bounded converging domains. We establish equivalent integral representations and provide strict two-sided estimates for their components. Then we provide a two-sided estimate for the Lebesgue constant. We solve completely the problem on the upper bound of the constant $L_n$. We improve its known lower bound.

Keywords: partial sums of Fourier series, norm of Fourier operator, Lebesgue constant, asymptotic formula, estimate for Lebesgue constant, extremal problem.

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English version:
Ufa Mathematical Journal, 2018, 10:1, 94–114 (PDF, 408 kB); https://doi.org/10.13108/2018-10-1-94

Bibliographic databases:

Document Type: Article
UDC: 517.518.83
MSC: 34A25, 22E05
Received: 14.07.2016

Citation: I. A. Shakirov, “On two-sided estimate for norm of Fourier operator”, Ufimsk. Mat. Zh., 10:1 (2018), 96–117; Ufa Math. J., 10:1 (2018), 94–114

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