Wiener Tauberian theorem

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

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

See also Tauberian theorems.

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)