Wiener Tauberian theorem
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) |
Wiener Tauberian theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_Tauberian_theorem&oldid=16501