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.

7 pages

References in corpus (2)

• Quantified versions of Ingham's theorem
• A quantified Tauberian theorem for sequences

Cited by in corpus (6)

• Semi-uniform stability of operator semigroups and energy decay of damped waves
• An abstract approach to optimal decay of functions and operator semigroups
• On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approach
• 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