Abstract
A class of permutations of the Walsh-Paley system that preserve the lebesgue constants and theL p-norms of the Dirichlet kernel is distinguished. Thus it is proved, in particular, that Fine’s estimates and calculations of the Lebesgue constants for the Walsh—Paley system hold for the Walsh systems in the enumerations of Walsh and Kaczmarz. A third algorithm for calculating the Lebesgue constants, which is different from those obtained by Fine and which also makes it possible to calculate theL p-norms of the Dirichlet kernels, is proposed. It is shown that not all permutations of the Walsh system even within the blocks preserve the Lebesgue constants. The distinguished class of permutations includes theTW-systems of Schipp, which are not, in general, permutations within the blocks.
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Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 36–48, July, 2000.
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Bespalov, M.S. Permutations of the Walsh system that preserve Lebesgue constants. Math Notes 68, 32–42 (2000). https://doi.org/10.1007/BF02674643
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DOI: https://doi.org/10.1007/BF02674643