Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T23:32:27.697Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

Extract

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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