Asymptotic Behavior of Trigonometric Series with O-Regularly Varying Quasimonotone Coefficients

Abstract

In this paper, the asymptotic behavior of a trigonometric series with O-regularly varying quasimonotone coefficients is investigated. Our results generalize the work Ž Ž . done by S. Aljancic, R. Bojanic, and M.

Trigonometric series with general monotone coefficients

“…On the other hand (see [ We note that the previous results generalize many results of this type starting with the paper of Hardy [10]; see, for example, the papers [21] for QM case [7,31,32] for ORVQM case. The next theorem extends the results of [8].…”

“…The convergence result does not hold for a series with O-regularly varying quasi-monotone coefficients (see [7]). So, by 5 • , it does not hold and for the series with general monotone coefficients.…”

Hardy Type Asymptotics for Cosine Series in Several Variables with Decreasing Power-Like Coefficients

“…This indeterminate form can be resolved also with the help of the identity Γ(1 − α) cos( πα 2 ) ≡ π 2Γ(α) sin( πα 2 ) . Note that in works [1][2][3][4][5], as well as in subsequent publications [6][7][8][9][10], one can find quite a number of deeper results than those mentioned earlier, part of which are included in monograph [11,Ch. V].…”

“…It is not clear whether it is possible to apply, in the case d ≥ 2, the methods developed in [1][2][3][4][5][6][7][8][9][10] for studying the asymptotics of trigonometric series due to their one-dimensional specificity. Therefore in our case we use the approach of "reduction to dimension one" allowing to express the function F d (θ) as an explicit combination of the one-dimensional functions F 1 (·), or rather of the functions H α (·).…”

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

“…The asymptotics of trigonometric series with quasimonotone coefficients was studied in [6][7][8][9]. In the context of the indicated problem, in [10,11], quasimonotonicity and extensions of regularly varying and slowly varying coefficients were considered. Note that it is possible to improve the asymptotics for the last case using the results of the present paper.…”