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Wiener Tauberian theorem

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If has nowhere vanishing Fourier transform and is a function in such that the convolution tends to zero as , then the convolution , for any tends to zero as . Established by N. Wiener [1]. This theorem was generalized to include any commutative locally compact non-compact group : If is a function on , summable with respect to the Haar measure, whose Fourier transform does not vanish on the group of characters of and if is a function in such that the convolution tends to zero at infinity on , then the convolution tends to zero at infinity on for all summable functions on .

This theorem is based on the regularity of the group algebra of a commutative locally compact group, and on the possibility of spectral synthesis in group algebras for closed ideals belonging to only a finite number of regular maximal ideals [3].

References

[1] N. Wiener, "Tauberian theorems" Ann. of Math. (2) , 33 : 1 (1932) pp. 1–100
[2] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[3] N. Bourbaki, "Théories spectrales" , Eléments de mathématiques , Hermann (1967)


Comments

References

[a1] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1970)
[a2] W. Rudin, "Fourier analysis on groups" , Interscience (1962)
[a3] H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)
How to Cite This Entry:
Wiener Tauberian theorem. A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Wiener_Tauberian_theorem&oldid=16501
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098