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Uniform with Respect to the Parameter a∈(0,1) Two-Sided Estimates of the Sums of Sine and Cosine Series with Coefficients 1/ka by the First Terms of Their Asymptoticsстатья Исследовательская статья
Статья опубликована в журнале из списка Web of Science и/или Scopus
- Авторы: Popov A.Y., Rodionov T.V.
- Журнал: Proceedings of the Steklov Institute of Mathematics
- Том: 319
- Номер: Suppl.1
- Год издания: 2022
- Издательство: Springer Verlag
- Местоположение издательства: Germany
- Первая страница: S204
- Последняя страница: S217
- DOI: 10.1134/S0081543822060189
- Аннотация: Uniform with respect to the parameter a∈(0,1) estimates of the functions fa(x)=∑∞k=1k−acoskx and ga(x)=∑∞k=1k−asinkx by the first terms of their asymptotic expansions Fa(x)=sin(πa/2)Γ(1−a)xa−1 and Ga(x)=cos(πa/2)Γ(1−a)xa−1 are obtained. Namely, it is proved that the inequalities Ga(x)−x2<ga(x)<Ga(x)−x12, Fa(x)+ζ(a)+ζ(3)4π3x2sin(πa/2)<fa(x)<Fa(x)+ζ(a)+118x2sin(πa/2) are valid for all a∈(0,1) and x∈(0,π]. It is shown that the estimates are unimprovable in the following sense. In the lower estimate for the sine series, the subtrahend x/2 cannot be replaced by kx with any k<1/2: the estimate ceases to be fulfilled for sufficiently small x and the values of a close to 1. In the upper estimate, the subtrahend x/12 cannot be replaced by kx with any k>1/12: the estimate ceases to be fulfilled for the values of a and x close to 0. In the lower estimate for the cosine series, the multiplier ζ(3)/(4π3) of x2sin(πa/2) cannot be replaced by any larger number: the estimate ceases to be fulfilled for x and a close to 0. In the upper estimate for the cosine series, the multiplier 1/18 of x2sin(πa/2) can probably be replaced by a smaller number but not by 1/24: for every a∈[0.98,1), such an estimate would not hold at the point x=π as well as on a certain closed interval x0(a)≤x≤π, where x0(a)→0 as a→1−. The obtained results allow us to refine the estimates for the functions fa and ga established recently by other authors.
- Добавил в систему: Родионов Тимофей Викторович