A Wiener type theorem for Dirichlet series
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- by Arthur Goodman and D. J. Newman PDF
- Proc. Amer. Math. Soc. 92 (1984), 521-527 Request permission
Abstract:
A famous theorem of Wiener states the conditions under which the reciprocal of a function with an absolutely convergent Fourier series also has an absolutely convergent Fourier series. We offer an elementary proof of the fact, first proven in [2], that if $F(s)$ has an absolutely convergent Dirichlet series then $1/F(s)$ has an absolutely convergent Dirichlet series if and only if $\left | {F(s)} \right |$ is bounded away from zero in the closed right half-plane.References
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G. H. Hardy and M. Riesz, The general theory of Dirichlet series, Cambridge Univ. Press, 1952.
- D. A. Edwards, On absolutely convergent Dirichlet series, Proc. Amer. Math. Soc. 8 (1957), 1067–1074. MR 96086, DOI 10.1090/S0002-9939-1957-0096086-1
- Edwin Hewitt and J. H. Williamson, Note on absolutely convergent Dirichlet series, Proc. Amer. Math. Soc. 8 (1957), 863–868. MR 90680, DOI 10.1090/S0002-9939-1957-0090680-X L. Hörmander, An introduction to complex analysis in several complex variables, Van Nostrand, 1966.
- D. J. Newman, A simple proof of Wiener’s $1/f$ theorem, Proc. Amer. Math. Soc. 48 (1975), 264–265. MR 365002, DOI 10.1090/S0002-9939-1975-0365002-8
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 521-527
- MSC: Primary 30B50; Secondary 32A99, 46H99
- DOI: https://doi.org/10.1090/S0002-9939-1984-0760938-3
- MathSciNet review: 760938