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An integral transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w0971201.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w0971202.png" />, defined as follows:
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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/w/w097/w097120/w0971203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w0971204.png" /> is a real variable, the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w0971205.png" /> has the form
+
An integral transform  $  g $
 +
of a function  $  f \in {L _ {2} } ( 0, \infty ) $,
 +
defined as follows:
  
<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/w/w097/w097120/w0971206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
g( x)  =
 +
\frac{d}{dx}
 +
\int\limits _ { 0 } ^  \infty  \widetilde \omega  ( xu) f( u)
 +
\frac{du}{u}
 +
.
 +
$$
  
(l.i.m. denotes the limit in the mean in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w0971207.png" />) and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w0971208.png" /> satisfies the condition
+
Here  $  x $
 +
is a real variable, the kernel  $  {\widetilde \omega  } ( x) $
 +
has the form
  
<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/w/w097/w097120/w0971209.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
\widetilde \omega  ( x)  =
 +
\frac{x}{2 \pi }
  
The following conditions are sufficient for the existence of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712010.png" /> and the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712011.png" />:
+
\mathop{\rm l}.i.m. _ {T \rightarrow \infty } \
 +
\int\limits _ { - } T ^ { T } 
 +
\frac{\Omega \left (
 +
\frac{1}{2}
 +
+ it \right ) }{
 +
\frac{1}{2}
 +
- it }
 +
x ^ {- ( t+ 1/2) }  dt
 +
$$
  
<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/w/w097/w097120/w09712012.png" /></td> </tr></table>
+
(l.i.m. denotes the limit in the mean in  $  L _ {2} $)
 +
and the function  $  \Omega ( it + 1 / 2) $
 +
satisfies the condition
 +
 
 +
$$
 +
\Omega ( s) \Omega ( 1- s)  =  1.
 +
$$
 +
 
 +
The following conditions are sufficient for the existence of the kernel  $  {\widetilde \omega  } ( x) $
 +
and the inclusion  $  {\widetilde \omega  ( x) } / x \in {L _ {2} } ( 0, \infty ) $:
 +
 
 +
$$
 +
\Omega \left (
 +
\frac{1}{2}
 +
- it \right )  =  \Omega \left (
 +
\frac{1}{2}
 +
+ it \right )
 +
$$
  
 
and
 
and
  
<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/w/w097/w097120/w09712013.png" /></td> </tr></table>
+
$$
  
For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712014.png" />, formula (1) defines the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712015.png" /> almost-everywhere. The inversion formula for the Watson transform (1) has the form
+
\frac{\Omega \left (  
 +
\frac{1}{2}
 +
+ it \right ) }{
 +
\frac{1}{2}
 +
- it }
 +
  \in \
 +
L _ {2} (- \infty , \infty ).
 +
$$
  
<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/w/w097/w097120/w09712016.png" /></td> </tr></table>
+
For a function  $  f \in L _ {2} ( 0, \infty ) $,
 +
formula (1) defines the function  $  g \in L _ {2} ( 0, \infty ) $
 +
almost-everywhere. The inversion formula for the Watson transform (1) has the form
 +
 
 +
$$
 +
f( x)  =
 +
\frac{d}{dx}
 +
\int\limits _ { 0 } ^  \infty  \widetilde \omega  ( xu )
 +
g( u) 
 +
\frac{du}{u}
 +
.
 +
$$
  
 
Named after G.N. Watson [[#References|[1]]], who was the first to study this transform.
 
Named after G.N. Watson [[#References|[1]]], who was the first to study this transform.
Line 28: Line 91:
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.N. Watson,  "General transforms"  ''Proc. London Math. Soc. (2)'' , '''35'''  (1933)  pp. 156–199</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.N. Watson,  "General transforms"  ''Proc. London Math. Soc. (2)'' , '''35'''  (1933)  pp. 156–199</TD></TR></table>
  
 +
====Comments====
 +
Quite generally, let  $  \psi $
 +
be a Lebesgue-measure function in  $  \mathbf R _ {>} 0 $
 +
and let
  
 +
$$
 +
\phi  =  \int\limits _ { 0 } ^ { x }  \psi ( x)  dt .
 +
$$
  
====Comments====
+
The kernel  $  \psi $(
Quite generally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712017.png" /> be a Lebesgue-measure function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712018.png" /> and let
+
or  $  \phi $)
 
+
is called a generalized kernel, or kernel of a generalized transform, if
<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/w/w097/w097120/w09712019.png" /></td> </tr></table>
 
  
The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712020.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712021.png" />) is called a generalized kernel, or kernel of a generalized transform, if
+
a)  $  \psi ( x) $
 +
is real valued on  $  \mathbf R _ {>} 0 $;
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712022.png" /> is real valued on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712023.png" />;
+
b) $  x  ^ {-} 1 \phi ( x) \in L _ {2} ( \mathbf R _ {>} 0 ) $;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712024.png" />;
+
c) $  \int _ {0}  ^  \infty  \phi ( xu ) \phi ( yu ) u  ^ {-} 2  du = \min ( x, y) $.
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712025.png" />.
+
The operator  $  \Phi $
 +
defined on  $  L _ {2} ( \mathbf R _ {>} 0 ) $
 +
by
  
The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712026.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097120/w09712027.png" /> by
+
$$
 +
\Phi ( f  )( x)  =
 +
\frac{d}{dx}
 +
\int\limits _ { 0 } ^  \infty 
  
<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/w/w097/w097120/w09712028.png" /></td> </tr></table>
+
\frac{\phi ( xt) f( t) }{t}
 +
  dt
 +
$$
  
 
is called a generalized transform or Watson transform.
 
is called a generalized transform or Watson transform.

Revision as of 08:28, 6 June 2020


An integral transform $ g $ of a function $ f \in {L _ {2} } ( 0, \infty ) $, defined as follows:

$$ \tag{1 } g( x) = \frac{d}{dx} \int\limits _ { 0 } ^ \infty \widetilde \omega ( xu) f( u) \frac{du}{u} . $$

Here $ x $ is a real variable, the kernel $ {\widetilde \omega } ( x) $ has the form

$$ \tag{2 } \widetilde \omega ( x) = \frac{x}{2 \pi } \mathop{\rm l}.i.m. _ {T \rightarrow \infty } \ \int\limits _ { - } T ^ { T } \frac{\Omega \left ( \frac{1}{2} + it \right ) }{ \frac{1}{2} - it } x ^ {- ( t+ 1/2) } dt $$

(l.i.m. denotes the limit in the mean in $ L _ {2} $) and the function $ \Omega ( it + 1 / 2) $ satisfies the condition

$$ \Omega ( s) \Omega ( 1- s) = 1. $$

The following conditions are sufficient for the existence of the kernel $ {\widetilde \omega } ( x) $ and the inclusion $ {\widetilde \omega ( x) } / x \in {L _ {2} } ( 0, \infty ) $:

$$ \Omega \left ( \frac{1}{2} - it \right ) = \Omega \left ( \frac{1}{2} + it \right ) $$

and

$$ \frac{\Omega \left ( \frac{1}{2} + it \right ) }{ \frac{1}{2} - it } \in \ L _ {2} (- \infty , \infty ). $$

For a function $ f \in L _ {2} ( 0, \infty ) $, formula (1) defines the function $ g \in L _ {2} ( 0, \infty ) $ almost-everywhere. The inversion formula for the Watson transform (1) has the form

$$ f( x) = \frac{d}{dx} \int\limits _ { 0 } ^ \infty \widetilde \omega ( xu ) g( u) \frac{du}{u} . $$

Named after G.N. Watson [1], who was the first to study this transform.

References

[1] G.N. Watson, "General transforms" Proc. London Math. Soc. (2) , 35 (1933) pp. 156–199

Comments

Quite generally, let $ \psi $ be a Lebesgue-measure function in $ \mathbf R _ {>} 0 $ and let

$$ \phi = \int\limits _ { 0 } ^ { x } \psi ( x) dt . $$

The kernel $ \psi $( or $ \phi $) is called a generalized kernel, or kernel of a generalized transform, if

a) $ \psi ( x) $ is real valued on $ \mathbf R _ {>} 0 $;

b) $ x ^ {-} 1 \phi ( x) \in L _ {2} ( \mathbf R _ {>} 0 ) $;

c) $ \int _ {0} ^ \infty \phi ( xu ) \phi ( yu ) u ^ {-} 2 du = \min ( x, y) $.

The operator $ \Phi $ defined on $ L _ {2} ( \mathbf R _ {>} 0 ) $ by

$$ \Phi ( f )( x) = \frac{d}{dx} \int\limits _ { 0 } ^ \infty \frac{\phi ( xt) f( t) }{t} dt $$

is called a generalized transform or Watson transform.

References

[a1] G.O. Okikiolu, "Aspects of the theory of bounded operators in -spaces" , Acad. Press (1971) pp. §6.7
How to Cite This Entry:
Watson transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Watson_transform&oldid=12261
This article was adapted from an original article by A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article