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).

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…Observe, however, that Theorem 3.1 is best possible in the sense that one cannot expect any rate of decay for f if no further assumptions are imposed on the growth of ffalse^, even if ffalse^ extends to an entire function; see e.g. [69,74,75]. In order to obtain quantitative results, one assumes that ffalse^ extends analytically beyond iR to some precise domain and that this analytic extension satisfies an appropriate bound in this domain.…”

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…Observe, however, that Theorem 3.1 is best possible in the sense that one cannot expect any rate of decay for f if no further assumptions are imposed on the growth of f , even if f extends to an entire function; see e.g. [33,54,55]. In order to obtain quantitative results one assumes that f extends analytically beyond iR to some precise domain and that this analytic extension satisfies an appropriate bound in this domain.…”