View source for Wiener Tauberian theorem

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979501.png" /> has nowhere vanishing [[Fourier transform|Fourier transform]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979502.png" /> is a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979503.png" /> such that the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979504.png" /> tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979505.png" />, then the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979506.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979507.png" /> tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979508.png" />. Established by N. Wiener [[#References|[1]]]. This theorem was generalized to include any commutative locally compact non-compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w0979509.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795010.png" /> is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795011.png" />, summable with respect to the [[Haar measure|Haar measure]], whose Fourier transform does not vanish on the group of characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795013.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795014.png" /> is a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795015.png" /> such that the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795016.png" /> tends to zero at infinity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795017.png" />, then the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795018.png" /> tends to zero at infinity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795019.png" /> for all summable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097950/w09795021.png" />.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Wiener,   "Tauberian theorems"  ''Ann. of Math. (2)'' , '''33''' :  1  (1932)  pp. 1–100</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Théories spectrales" , ''Eléments de mathématiques'' , Hermann  (1967)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''2''' , Springer  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,   "Fourier analysis on groups" , Interscience  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Reiter,   "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR></table>