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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" />.
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{{TEX|done}}
  
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]]].
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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 [[#References|[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 $G$.
 +
 
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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 [[#References|[3]]].
  
 
====References====
 
====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>
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<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====
 
====Comments====
 
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See also [[Tauberian theorems]].
  
 
====References====
 
====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>
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<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>
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</table>

Latest revision as of 18:46, 13 April 2017


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)
How to Cite This Entry:
Wiener Tauberian theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_Tauberian_theorem&oldid=40989
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article