Tauberian theorems for the LaplaceStieltjes transform
Author:
C. J. K. Batty
Journal:
Trans. Amer. Math. Soc. 322 (1990), 783804
MSC:
Primary 44A10; Secondary 30B50, 40E05, 47A60
MathSciNet review:
1013326
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Abstract 
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Additional Information
Abstract: Let be a function of locally bounded variation, with , whose LaplaceStieltjes transform is absolutely convergent for . Let be the singular set of in , and suppose that . Various estimates for are obtained. In particular, as if This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.
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 [1]
 G. R. Allan, A. G. O'Farrell, and T. J. Ransford, A Tauberian theorem arising in operator theory, Bull. London Math. Soc. 19 (1987), 537545. MR 915430 (89c:47003)
 [2]
 W. Arendt and C. J. K. Batty, Tauberian theorems and stability of oneparameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837852. MR 933321 (89g:47053)
 [3]
 A. E. Ingham, On Wiener's method in Tauberian theorems, Proc. London Math. Soc. (2) 38 (1935), 458480.
 [4]
 Y. Katznelson and L. Tzafriri, On power bounded operators, J. Funct. Anal. 68 (1986), 313328. MR 859138 (88e:47006)
 [5]
 J. Korevaar, On Newman's quick way to the prime number theorem, Math. Intelligencer 4 (1982), 108115. MR 684025 (84b:10063)
 [6]
 Y. I. Lyubich and Vu Quoc Phong, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 3742. MR 932004 (89e:47062)
 [7]
 D. J. Newman, Simple analytic proof of the prime number theorem, Amer. Math. Monthly 87 (1980), 693696. MR 602825 (82h:10056)
 [8]
 T. J. Ransford, Some quantitative Tauberian theorems for power series, Bull. London Math. Soc. 20 (1988), 3744. MR 916072 (89g:30005)
 [9]
 E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, Oxford, 1932.
 [10]
 D. V. Widder, An introduction to transform theory, Academic Press, New York, 1971.
 [11]
 D. Zagier, Short proof of the prime number theorem, unpublished manuscript.
 [12]
 C. J. K. Batty and Vu Quoc Phong, Stability of individual elements under oneparameter semigroups, Trans. Amer. Math. Soc. 322 (1990), 805818. MR 1022866 (91c:47072)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199010133266
PII:
S 00029947(1990)10133266
Keywords:
Tauberian theorem,
LaplaceStieltjes transform,
Laplace transform,
power series,
Dirichlet series,
semigroup
Article copyright:
© Copyright 1990 American Mathematical Society
