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Integrability of Fourier transforms under an ergodic hypothesis

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Gavin Brown
Gavin Brown
Affiliation:
School of Mathematics University of New South Wales P.O. Box 1 Kensington NSW 2033 Australia
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Abstract

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The object is to unify and complement some recent theorems of Hewitt and Ritter on the integrability of Fourier transforms, but the underlying theme is the ancient one that Plancherel's theorem is the “only” integrability constraint on Fourier transforms. The distinguishing feature of the results is that we restrict attention to positive measures (or functions) which satisfy an ergodic condition and whose transforms are positive. (In fact we employ sums of discrete random variables, a technique which seems to have been largely ignored in context.) The general setting is that of locally compact abelian groups but we are chiefly interested in the line or the circle, and it appears that the theorems are new for these classical groups.

Keywords

Fourier-Stieltjes transformsdiscrete random variables.

MSC classification

Secondary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 2 , September 1978 , pp. 129 - 153
Copyright
Copyright © Australian Mathematical Society 1978

References

Boas, R. P. Jnr, (1967), Integrability Theorems for Trigonometric Transforms (Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
Brown, G. (1973), “M 0(G) has a symmetric maximal ideal off the Šilov boundary”, Proc. London Math. Soc. 27, 484–504.CrossRefGoogle Scholar
Brown, G. (1975), “Riesz products and generalized characters”, Proc. London Math. Soc. 30, 209–238.CrossRefGoogle Scholar
Brown, G. (1977), “Singular infinitely divisible distributions whose characteristic functions vanish at infinity”, Math. Proc. Camb. Phil.Soc. 82, 277–287.CrossRefGoogle Scholar
Brown, G. (1977), “Fourier transforms on the line and Gronwall's theorem”, J. London Math. Soc. (to appear).Google Scholar
Brown, G. and Moran, W. (1973), “A dichotomy for infinite convolutions of discrete measures”, Proc. Camb. Phil. Soc. 73, 307–316.CrossRefGoogle Scholar
Brown, G. and Moran, W. (1974), “Products of random variables and Kakutani's criterion for orthogonality of products measures”, J. London Math. Soc. 76, 173–181.Google Scholar
Brown, G. and Moran, W. (1975), “Products of random variables and Kakutani's criterion for orthogonality of product measures”, J. London Math. Soc. 10, 401–405.CrossRefGoogle Scholar
Brown, G. and Moran, W. (1975), “Coin tossing and powers of singular measures”, Math. Proc. Camb. Phil. Soc. 77, 349–364.CrossRefGoogle Scholar
Gronwall, T. H. (1921), “On the Fourier coefficients of a continuous function”, Bull. Amer. Math.Soc. 27, 320–321.CrossRefGoogle Scholar
Hewitt, E. and Ritter, G. (1976), “Über die Integrierbarkeit von Fourier-Trasformierten auf Gruppen; Teil I: Stetige Funktionen mit kompaktem Träger und eine Bemerkung über hyperbolische Differentialoperatoren”, Math. Ann. 224, 77–96.CrossRefGoogle Scholar
Hewitt, E. and Ritter, G. (1977), “On the integrability of Fourier transforms on groups; Part II: Fourier-Stieltjes transforms of singular measures”, Proc. Roy.Irich Acad. 76A, 265–287.Google Scholar
Hewitt, E. and Zuckerman, H. S. (1967), “Singular measures with absolutely continuous squares”, Proc. Camb. Phil. Soc. 62, 399–420;CrossRefGoogle Scholar
corrigendum Proc. Camb. Phil. Soc. 63, 367–368.Google Scholar
Reiter, H. J. (1968), Classical Harmonic Analysis and Locally Compact Abelian Groups (Oxford University Press, 1968).Google Scholar
Saeki, S. (1977), “On infinite convolution products of discrete probability measures”, J. London Math. Soc. 16, 172–176.CrossRefGoogle Scholar
Salem, R. and Zygmund, A. (1947), “On a theorem of Banach”, Proc. Nat.Acad.Sci. U.S.A. 33, 293–295.CrossRefGoogle ScholarPubMed
Stromberg, K. (1968), “Large families of singular measures having absolutely continuous convolution squares”, Proc. Camb. Phil. Soc. 64, 1015–1022.CrossRefGoogle Scholar
Zygmund, A. (1959), Trigonometric Series, Vols. I and II (Cambridge University Press, 1959).Google Scholar