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The linear functional transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435601.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435602.png" /> defined by the integral
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435603.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435604.png" /></td> </tr></table>
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The linear functional transform  $  W ( \zeta ) [ x] $
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of a function  $  x( t) $
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defined by the integral
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435605.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435606.png" />; for real values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435607.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435608.png" /> is self-adjoint [[#References|[1]]]. The inversion formula for the Gauss transform is
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$$
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W ( \zeta ) [ x] = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g0435609.png" /></td> </tr></table>
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\frac{1}{\sqrt {\pi \zeta } }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g04356010.png" />, the Gauss transform is known as the Weierstrass transform.
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\int\limits _ {- \infty } ^  \infty 
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\mathop{\rm exp} \left ( -
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\frac{u  ^ {2} } \zeta
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\right )
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x ( t + u)  du,
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$$
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$$
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\mathop{\rm Re}  \zeta  >  0.
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$$
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If $  x( t) \in L _ {2} ( - \infty , \infty ) $,
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then  $  W ( \zeta ) [ x] \in L _ {2} ( - \infty , \infty ) $;
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for real values  $  \zeta = \overline \zeta \; $,
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the operator  $  W ( \zeta ) $
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is self-adjoint [[#References|[1]]]. The inversion formula for the Gauss transform is
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$$
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x ( t)  =   \mathop{\rm exp} \left \{ - {
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\frac \zeta {4}
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}
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\frac{d  ^ {2} }{dt  ^ {2} }
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\right \} W ( \zeta ) [ x ( t)].
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$$
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If  $  \zeta = 4 $,  
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the Gauss transform is known as the Weierstrass transform.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms"  ''Itogi Nauk. Ser. Mat. Mat. Anal.''  (1966)  pp. 7–82  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms"  ''Itogi Nauk. Ser. Mat. Mat. Anal.''  (1966)  pp. 7–82  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The above inversion formula can be interpreted in terms of semi-groups. Another way to invert the Gauss transform is to write in the first equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043560/g04356011.png" />, from which substitution a double-sided [[Laplace transform|Laplace transform]] results. Then the inversion formula follows from well-known Laplace-transform techniques.
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The above inversion formula can be interpreted in terms of semi-groups. Another way to invert the Gauss transform is to write in the first equation $  t + u = v $,  
 +
from which substitution a double-sided [[Laplace transform|Laplace transform]] results. Then the inversion formula follows from well-known Laplace-transform techniques.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)</TD></TR></table>

Latest revision as of 19:41, 5 June 2020


The linear functional transform $ W ( \zeta ) [ x] $ of a function $ x( t) $ defined by the integral

$$ W ( \zeta ) [ x] = \ \frac{1}{\sqrt {\pi \zeta } } \int\limits _ {- \infty } ^ \infty \mathop{\rm exp} \left ( - \frac{u ^ {2} } \zeta \right ) x ( t + u) du, $$

$$ \mathop{\rm Re} \zeta > 0. $$

If $ x( t) \in L _ {2} ( - \infty , \infty ) $, then $ W ( \zeta ) [ x] \in L _ {2} ( - \infty , \infty ) $; for real values $ \zeta = \overline \zeta \; $, the operator $ W ( \zeta ) $ is self-adjoint [1]. The inversion formula for the Gauss transform is

$$ x ( t) = \mathop{\rm exp} \left \{ - { \frac \zeta {4} } \frac{d ^ {2} }{dt ^ {2} } \right \} W ( \zeta ) [ x ( t)]. $$

If $ \zeta = 4 $, the Gauss transform is known as the Weierstrass transform.

References

[1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2] V.A. Ditkin, A.P. Prudnikov, "Integral transforms" Itogi Nauk. Ser. Mat. Mat. Anal. (1966) pp. 7–82 (In Russian)

Comments

The above inversion formula can be interpreted in terms of semi-groups. Another way to invert the Gauss transform is to write in the first equation $ t + u = v $, from which substitution a double-sided Laplace transform results. Then the inversion formula follows from well-known Laplace-transform techniques.

References

[a1] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)
How to Cite This Entry:
Gauss transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_transform&oldid=47053
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article