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Note on a theorem of Wiener and Pitt

Published online by Cambridge University Press:  26 February 2010

Dang Dinh Ang
Dang Dinh Ang
Affiliation:
57 Duy-Tan, Saigon, Vietnam.
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Let G be a locally compact Abelian non-discrete group. Let M(G) be the convolution algebra of Radon measures on G. Let µ be an element of M(G) with its Lebesgue decomposition [1]

into absolutely continuous, purely discontinuous and continuous singular parts. The chief problem one encounters in the study of the invertibility of µ is with the case µs ≠ 0. As observed by Wiener and Pitt [2], the problem can be handled provided µs be “not too large”. In fact, Wiener and Pitt (loc. cit.) proved the following:

Let µ be a Radon measure on R (the real line) such that

whereare the Fourier transforms of µ, µd, and ≑µs≑ is the variational norm of µs. Then, µ has an inverse in M (R).

Type
Research Article
Information
Mathematika , Volume 17 , Issue 2 , December 1970 , pp. 189 - 192
Copyright
Copyright © University College London 1970

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References

1.Hewitt, E. and Ross, K. A., Abstract harmonic analysis (Academic Press, 1963), 273.Google Scholar
2.Wiener, N. and Pitt, H. R., “Absolutely convergent Fourier transforms”, Duke Math. J., 4 (1938), 420–436.CrossRefGoogle Scholar
3.Varopoulos, N. Th., “Studies in harmonic analysis”, Proc. Camb. Phil. Soc., 60 (1964).CrossRefGoogle Scholar
4.Raikov, Gelfand and Silov, , Commutative normed rings (Chelsea, N.Y., 1964), §22.Google Scholar