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.
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G. Debruyne gratefully acknowledges support by Ghent University, through a BOF Ph.D. grant
The work of J. Vindas was supported by the Research Foundation–Flanders, through the FWO-grant number 1520515N
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Debruyne, G., Vindas, J. Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. JAMA 138, 799–833 (2019). https://doi.org/10.1007/s11854-019-0045-3
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DOI: https://doi.org/10.1007/s11854-019-0045-3