Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

Abstract

We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.

30 pages

References in corpus (2)

• On PNT equivalences for Beurling numbers
• A quantified Tauberian theorem for sequences

Cited by in corpus (5)

• Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth
• 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
• On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem
• $M(x)=o(x)$ Estimates for Beurling numbers