Geodesic
On two-sided estimate for norm of Fourier operator
Ufa mathematical journal, Tome 10 (2018) no. 1, pp. 94-114

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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, asymptotic formula, extremal problem.
Mots-clés : Lebesgue constant, estimate for Lebesgue constant 
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     author = {I. A. Shakirov},
     title = {On two-sided estimate  for norm of {Fourier} operator},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2018_10_1_a7/}
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I. A. Shakirov. On two-sided estimate  for norm of Fourier operator. Ufa mathematical journal, Tome 10 (2018) no. 1, pp. 94-114. http://geodesic.mathdoc.fr/item/UFA_2018_10_1_a7/