A Wiener type theorem for Dirichlet series

Authors:
Arthur Goodman and D. J. Newman

Journal:
Proc. Amer. Math. Soc. **92** (1984), 521-527

MSC:
Primary 30B50; Secondary 32A99, 46H99

MathSciNet review:
760938

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Abstract | References | Similar Articles | Additional Information

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 has an absolutely convergent Dirichlet series then has an absolutely convergent Dirichlet series if and only if is bounded away from zero in the closed right half-plane.

**[1]**G. H. Hardy and M. Riesz,*The general theory of Dirichlet series*, Cambridge Univ. Press, 1952.**[2]**D. A. Edwards,*On absolutely convergent Dirichlet series*, Proc. Amer. Math. Soc.**8**(1957), 1067–1074. MR**0096086**, 10.1090/S0002-9939-1957-0096086-1**[3]**Edwin Hewitt and J. H. Williamson,*Note on absolutely convergent Dirichlet series*, Proc. Amer. Math. Soc.**8**(1957), 863–868. MR**0090680**, 10.1090/S0002-9939-1957-0090680-X**[4]**L. Hörmander,*An introduction to complex analysis in several complex variables*, Van Nostrand, 1966.**[5]**D. J. Newman,*A simple proof of Wiener’s 1/𝑓 theorem*, Proc. Amer. Math. Soc.**48**(1975), 264–265. MR**0365002**, 10.1090/S0002-9939-1975-0365002-8

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0760938-3

Keywords:
Dirichlet series,
Banach algebras,
functions of several complex variables,
positive basis,
Kronecker's theorem,
Reinhardt domain

Article copyright:
© Copyright 1984
American Mathematical Society