Note on the absence of remainders in the Wiener-Ikehara theorem

Abstract

We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.2010 Mathematics Subject Classification. 11M45, 40E05, 44A10.

An abstract approach to optimal decay of functions and operator semigroups

“…It is based on a surprising application of the open mapping theorem inspired by [12] (where the idea is attributed to Hörmander). We mention that in [10] the open mapping theorem is used to prove another kind of optimality result for the Ingham-Karamata theorem; for two other related applications of the Baire category theorem and its consequences see [11,17].…”

An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions

“…In [16], it was conjectured that any non-negative non-decreasing function S, whose Mellin transform 3 +ε ), for each ε > 0. This conjecture was already disproved in [10]. In fact, in that paper it was even shown that for each positive function δ(x) → 0 there are functions S satisfying all requirements, yet (S(x) − Ax)/(xδ(x)) is unbounded.…”

“…We also mention that it is possible to give a constructive proof of the more general negative result from [10] that holds for any function δ(x) → 0; see our forthcoming article [4]. Let us now return to the main subject of our paper, the Fourier-Laplace transforms F α,β .…”

On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: A constructive approach

“…In the case of the Wiener-Ikehara theorem, this disproves a conjecture by Müger [12], who had conjectured that a certain remainder could be obtained if (1.1) can be analytically extended to Re s > α with some 0 < α < 1. It has indeed been shown in [9] that no stronger remainder than the one in (1.2) can be achieved if this extra assumption is solely made together with the classical hypotheses.…”

“…In a recent article [9] the last two named authors have proved that, in general, it is impossible to improve the error terms of the asymptotic formulas (1.2) and (1.3) in the Wiener-Ikehara theorem and the Ingham-Karamata theorem if one just augments the assumptions of these theorems by asking an additional analytic continuation hypothesis to a half-plane containing Re s > 1 or Re s > 0, respectively. In the case of the Wiener-Ikehara theorem, this disproves a conjecture by Müger [12], who had conjectured that a certain remainder could be obtained if (1.1) can be analytically extended to Re s > α with some 0 < α < 1.…”