Asymptotics of the Sum of a Sine Series with a Convex Slowly Varying Sequence of Coefficients

Aleksei Solodov
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Aleksei Solodov: Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, 119991 Moscow, Russia

Mathematics, 2021, vol. 9, issue 18, 1-12

Abstract: We study the asymptotic behavior in a neighborhood of zero of the sum of a sine series g ( b , x ) = ? k = 1 ? b k sin k x whose coefficients constitute a convex slowly varying sequence b . The main term of the asymptotics of the sum of such a series was obtained by Aljan?i?, Bojani?, and Tomi?. To estimate the deviation of g ( b , x ) from the main term of its asymptotics b m ( x ) / x , m ( x ) = [ ? / x ] , Telyakovski? used the piecewise-continuous function ? ( b , x ) = x ? k = 1 m ( x ) ? 1 k 2 ( b k ? b k + 1 ) . He showed that the difference g ( b , x ) ? b m ( x ) / x in some neighborhood of zero admits a two-sided estimate in terms of the function ? ( b , x ) with absolute constants independent of b . Earlier, the author found the sharp values of these constants. In the present paper, the asymptotics of the function g ( b , x ) on the class of convex slowly varying sequences in the regular case is obtained.

Keywords: sine series with monotone coefficients; convex sequence; slowly varying function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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