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

Abstract

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).

arXiv comment8 pages

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Cited by in corpus (3)

• An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions
• Optimal decay of semi-uniformly stable operator semigroups with empty spectrum
• On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem