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A century of complex Tauberian theory. (English) Zbl 1001.40007

Summary: Complex-analytic and related boundary properties of transforms give information on the behavior of pre-images. The transforms may be power series, Dirichlet series or Laplace-type integrals; the pre-images are series (of numbers) or functions.
The chief impulse for complex Tauberian theory came from number theory. The first part of the survey emphasizes methods which permit simple derivations of the prime number theorem, associated with the labels Landau-Wiener-Ikehara and Newman. Other important areas in complex Tauberian theory are associated with the names Fatou-Riesz and Ingham. Recent refinements have been motivated by operator theory and include local \(H^1\) and pseudofunction boundary behavior of transforms. Complex information has also led to better remainder estimates in connection with classical Tauberian theorems. Applications include the distribution of zeros and eigenvalues.

MSC:

40E05 Tauberian theorems
11M45 Tauberian theorems
30B50 Dirichlet series, exponential series and other series in one complex variable
44A10 Laplace transform
47A10 Spectrum, resolvent
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