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A transform of functions, having the form
 
A transform of functions, having 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/i/i051/i051680/i0516801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F ( x)  = \int\limits _ { C }
 +
K ( x , t ) f ( t) d t ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516802.png" /> is a finite or infinite contour in the complex plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516803.png" /> is the kernel of the integral transform (cf. [[Kernel of an integral operator|Kernel of an integral operator]]). In most cases one considers integral transforms for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516805.png" /> is the real axis or a part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516806.png" /> of it. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516807.png" />, then the transform is said to be finite. Formulas enabling one to recover the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516808.png" /> from a known <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i0516809.png" /> are called inversion formulas of the integral transform.
+
where $  C $
 +
is a finite or infinite contour in the complex plane and $  K ( x , t) $
 +
is the kernel of the integral transform (cf. [[Kernel of an integral operator|Kernel of an integral operator]]). In most cases one considers integral transforms for which $  K ( x , t ) \equiv K ( x t ) $
 +
and $  C $
 +
is the real axis or a part $  ( a , b ) $
 +
of it. If $  - \infty < a , b < \infty $,  
 +
then the transform is said to be finite. Formulas enabling one to recover the function $  f $
 +
from a known $  F $
 +
are called inversion formulas of the integral transform.
  
 
Examples of integral transforms. The Bochner transform:
 
Examples of integral transforms. The Bochner transform:
  
<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/i/i051/i051680/i05168010.png" /></td> </tr></table>
+
$$
 +
[ T f  ] ( r)  = 2 \pi r  ^ {1-} n/2
 +
\int\limits _ { 0 } ^  \infty 
 +
J _ {n/2-} 1 ( 2 \pi r \rho ) \rho  ^ {n/2} f ( \rho )  d \rho ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168011.png" /> is the Bessel function of the first kind of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168012.png" /> (cf. [[Bessel functions|Bessel functions]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168013.png" /> is the distance in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168014.png" />. The inversion formula is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168015.png" />. The Parseval identity is:
+
where $  J _  \nu  ( x) $
 +
is the Bessel function of the first kind of order $  \nu $(
 +
cf. [[Bessel functions|Bessel functions]]) and $  \rho $
 +
is the distance in $  \mathbf R  ^ {n} $.  
 +
The inversion formula is: $  f = T  ^ {2} f $.  
 +
The Parseval identity is:
  
<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/i/i051/i051680/i05168016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
| [ T f  ] ( r) |  ^ {2} r  ^ {k-} 1  d r  = \
 +
\int\limits _ { 0 } ^  \infty 
 +
| f ( \rho ) |  ^ {2} \rho  ^ {k-} 1  d \rho .
 +
$$
  
 
The Weber transform:
 
The Weber transform:
  
<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/i/i051/i051680/i05168017.png" /></td> </tr></table>
+
$$
 +
F ( u , a )  = \
 +
\int\limits _ { a } ^  \infty 
 +
c _  \nu  ( t u , a u ) t f ( t)  d t ,\  a \leq  t \leq  \infty ,
 +
$$
 +
 
 +
where  $  c _  \nu  ( \alpha , \beta ) \equiv J _  \nu  ( \alpha ) Y _  \nu  ( \beta ) - Y _  \nu  ( \alpha ) J _  \nu  ( \beta ) $
 +
and  $  J _  \nu  $
 +
and  $  Y _  \nu  $
 +
are the Bessel functions of first and second kind. The inversion formula is:
 +
 
 +
$$
 +
f ( x)  = \int\limits _ { 0 } ^  \infty 
 +
 
 +
\frac{c _  \nu  ( x u , a u ) }{J _  \nu  ^ {2} ( a u ) + Y _  \nu  ^ {2} ( a u ) }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168020.png" /> are the Bessel functions of first and second kind. The inversion formula is:
+
u F ( u , a )  d u .
 +
$$
  
<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/i/i051/i051680/i05168021.png" /></td> </tr></table>
+
For  $  a \rightarrow 0 $,
 +
the Weber transform turns into the Hankel transform:
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168022.png" />, the Weber transform turns into the Hankel transform:
+
$$
 +
F ( x)  = \int\limits _ { 0 } ^  \infty 
 +
\sqrt {x t } J _  \nu  ( x t ) f ( t)  d t ,\ \
 +
0 < x < \infty .
 +
$$
  
<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/i/i051/i051680/i05168023.png" /></td> </tr></table>
+
For  $  \nu = \pm  1/2 $
 +
this transform reduces to the Fourier sine and cosine transforms. The inversion formula is as follows: If  $  f \in L _ {1} ( 0 , \infty ) $,
 +
if  $  f $
 +
is of bounded variation in a neighbourhood of a point  $  t _ {0} > 0 $
 +
and if  $  \nu \geq  - 1/2 $,
 +
then
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168024.png" /> this transform reduces to the Fourier sine and cosine transforms. The inversion formula is as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168025.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168026.png" /> is of bounded variation in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168027.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168028.png" />, then
+
$$
  
<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/i/i051/i051680/i05168029.png" /></td> </tr></table>
+
\frac{f ( t _ {0} + 0 ) + f ( t _ {0} - 0 ) }{2}
 +
  = \
 +
\int\limits _ { 0 } ^  \infty 
 +
\sqrt {t _ {0} x } J _  \nu  ( t _ {0} x ) F ( x)  d x .
 +
$$
  
The Parseval identity: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168030.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168032.png" /> are the Hankel transforms of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168035.png" />, then
+
The Parseval identity: If $  \nu \geq  - 1/2 $,  
 +
if $  F $
 +
and $  G $
 +
are the Hankel transforms of the functions $  f $
 +
and $  g $,  
 +
where $  f , g \in L _ {1} ( 0 , \infty ) $,
 +
then
  
<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/i/i051/i051680/i05168036.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty  f ( t) g ( t)  d t  = \
 +
\int\limits _ { 0 } ^  \infty  F ( x) G ( x)  d x .
 +
$$
  
 
Other forms of the Hankel transform are:
 
Other forms of the Hankel transform are:
  
<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/i/i051/i051680/i05168037.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty  J _  \nu  ( x t ) t f ( t)  d t ,\ \
 +
\int\limits _ { 0 } ^  \infty 
 +
J _  \nu  ( 2 \sqrt {x t } ) f ( t)  d t .
 +
$$
  
 
The Weierstrass transform:
 
The Weierstrass transform:
  
<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/i/i051/i051680/i05168038.png" /></td> </tr></table>
+
$$
 +
f ( x)  = \
 +
 
 +
\frac{1}{\sqrt {4 \pi } }
 +
 
 +
\int\limits _ {- \infty } ^  \infty 
 +
\mathop{\rm exp}
 +
\left [ -
 +
\frac{( x - t )  ^ {2} }{4}
 +
\right ]
 +
f ( t )  d t ;
 +
$$
  
 
it is a special case of a [[Convolution transform|convolution transform]].
 
it is a special case of a [[Convolution transform|convolution transform]].
Line 45: Line 135:
 
Repeated transforms. Let
 
Repeated transforms. Let
  
<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/i/i051/i051680/i05168039.png" /></td> </tr></table>
+
$$
 +
f _ {i+} 1 ( x)  = \
 +
\int\limits _ { 0 } ^  \infty 
 +
f _ {i} ( t) K _ {i} ( x t )  d t ,\ \
 +
i = 1 \dots n ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168040.png" />. Such a sequence of integral transforms is called a chain of integral transforms. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168041.png" />, repeated integral transforms are often called Fourier transforms.
+
where $  f _ {n+} 1 ( x) = f _ {1} ( x) $.  
 +
Such a sequence of integral transforms is called a chain of integral transforms. For $  n = 2 $,  
 +
repeated integral transforms are often called Fourier transforms.
  
Multiple (multi-dimensional) integral transforms are transforms (1) where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168043.png" /> is some domain in the complex Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168044.png" />-dimensional space.
+
Multiple (multi-dimensional) integral transforms are transforms (1) where $  t , x \in \mathbf R  ^ {n} $
 +
and $  C $
 +
is some domain in the complex Euclidean $  n $-
 +
dimensional space.
  
 
Integral transforms of generalized functions can be constructed by the following basic methods:
 
Integral transforms of generalized functions can be constructed by the following basic methods:
  
1) One constructs a space of test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168045.png" /> containing the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168046.png" /> of the integral transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168047.png" /> under consideration. Then the transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168048.png" /> for any generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168049.png" /> is defined as the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168050.png" /> on the test function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168051.png" /> according to the formula
+
1) One constructs a space of test functions $  U $
 +
containing the kernel $  K ( x , t ) $
 +
of the integral transform $  T $
 +
under consideration. Then the transform $  T f $
 +
for any generalized function $  f \in U  ^  \prime  $
 +
is defined as the value of $  f $
 +
on the test function $  K ( x , t ) $
 +
according to the formula
  
<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/i/i051/i051680/i05168052.png" /></td> </tr></table>
+
$$
 +
T [ f ( t) ] ( x)  = \langle  f , K ( x , t ) \rangle .
 +
$$
  
2) A space of test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168053.png" /> is constructed on which the classical integral transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168054.png" /> is defined, mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168055.png" /> onto some space of test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168056.png" />. Then the integral transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168057.png" /> of a generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051680/i05168058.png" /> is defined by the equation
+
2) A space of test functions $  U $
 +
is constructed on which the classical integral transform $  T $
 +
is defined, mapping $  U $
 +
onto some space of test functions $  V $.  
 +
Then the integral transform $  T  ^  \prime  $
 +
of a generalized function $  f \in V  ^  \prime  $
 +
is defined by the equation
  
<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/i/i051/i051680/i05168059.png" /></td> </tr></table>
+
$$
 +
\langle  T  ^  \prime  f , \phi \rangle  = \langle  f , T \phi \rangle ,\ \
 +
\phi \in U .
 +
$$
  
 
3) The required integral transform is expressed in terms of another integral transform that is defined for generalized functions.
 
3) The required integral transform is expressed in terms of another integral transform that is defined for generalized functions.
Line 67: Line 185:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Operational calculus"  ''Progress in Math.'' , '''1'''  (1968)  pp. 1–75  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Y.A. Brychkov,  A.P. Prudnikov,  "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Operational calculus"  ''Progress in Math.'' , '''1'''  (1968)  pp. 1–75  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Y.A. Brychkov,  A.P. Prudnikov,  "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====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><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Zemanian,  "Generalized integral transformations" , Interscience  (1968)</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><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Zemanian,  "Generalized integral transformations" , Interscience  (1968)</TD></TR></table>

Revision as of 22:12, 5 June 2020


A transform of functions, having the form

$$ \tag{1 } F ( x) = \int\limits _ { C } K ( x , t ) f ( t) d t , $$

where $ C $ is a finite or infinite contour in the complex plane and $ K ( x , t) $ is the kernel of the integral transform (cf. Kernel of an integral operator). In most cases one considers integral transforms for which $ K ( x , t ) \equiv K ( x t ) $ and $ C $ is the real axis or a part $ ( a , b ) $ of it. If $ - \infty < a , b < \infty $, then the transform is said to be finite. Formulas enabling one to recover the function $ f $ from a known $ F $ are called inversion formulas of the integral transform.

Examples of integral transforms. The Bochner transform:

$$ [ T f ] ( r) = 2 \pi r ^ {1-} n/2 \int\limits _ { 0 } ^ \infty J _ {n/2-} 1 ( 2 \pi r \rho ) \rho ^ {n/2} f ( \rho ) d \rho , $$

where $ J _ \nu ( x) $ is the Bessel function of the first kind of order $ \nu $( cf. Bessel functions) and $ \rho $ is the distance in $ \mathbf R ^ {n} $. The inversion formula is: $ f = T ^ {2} f $. The Parseval identity is:

$$ \int\limits _ { 0 } ^ \infty | [ T f ] ( r) | ^ {2} r ^ {k-} 1 d r = \ \int\limits _ { 0 } ^ \infty | f ( \rho ) | ^ {2} \rho ^ {k-} 1 d \rho . $$

The Weber transform:

$$ F ( u , a ) = \ \int\limits _ { a } ^ \infty c _ \nu ( t u , a u ) t f ( t) d t ,\ a \leq t \leq \infty , $$

where $ c _ \nu ( \alpha , \beta ) \equiv J _ \nu ( \alpha ) Y _ \nu ( \beta ) - Y _ \nu ( \alpha ) J _ \nu ( \beta ) $ and $ J _ \nu $ and $ Y _ \nu $ are the Bessel functions of first and second kind. The inversion formula is:

$$ f ( x) = \int\limits _ { 0 } ^ \infty \frac{c _ \nu ( x u , a u ) }{J _ \nu ^ {2} ( a u ) + Y _ \nu ^ {2} ( a u ) } u F ( u , a ) d u . $$

For $ a \rightarrow 0 $, the Weber transform turns into the Hankel transform:

$$ F ( x) = \int\limits _ { 0 } ^ \infty \sqrt {x t } J _ \nu ( x t ) f ( t) d t ,\ \ 0 < x < \infty . $$

For $ \nu = \pm 1/2 $ this transform reduces to the Fourier sine and cosine transforms. The inversion formula is as follows: If $ f \in L _ {1} ( 0 , \infty ) $, if $ f $ is of bounded variation in a neighbourhood of a point $ t _ {0} > 0 $ and if $ \nu \geq - 1/2 $, then

$$ \frac{f ( t _ {0} + 0 ) + f ( t _ {0} - 0 ) }{2} = \ \int\limits _ { 0 } ^ \infty \sqrt {t _ {0} x } J _ \nu ( t _ {0} x ) F ( x) d x . $$

The Parseval identity: If $ \nu \geq - 1/2 $, if $ F $ and $ G $ are the Hankel transforms of the functions $ f $ and $ g $, where $ f , g \in L _ {1} ( 0 , \infty ) $, then

$$ \int\limits _ { 0 } ^ \infty f ( t) g ( t) d t = \ \int\limits _ { 0 } ^ \infty F ( x) G ( x) d x . $$

Other forms of the Hankel transform are:

$$ \int\limits _ { 0 } ^ \infty J _ \nu ( x t ) t f ( t) d t ,\ \ \int\limits _ { 0 } ^ \infty J _ \nu ( 2 \sqrt {x t } ) f ( t) d t . $$

The Weierstrass transform:

$$ f ( x) = \ \frac{1}{\sqrt {4 \pi } } \int\limits _ {- \infty } ^ \infty \mathop{\rm exp} \left [ - \frac{( x - t ) ^ {2} }{4} \right ] f ( t ) d t ; $$

it is a special case of a convolution transform.

Repeated transforms. Let

$$ f _ {i+} 1 ( x) = \ \int\limits _ { 0 } ^ \infty f _ {i} ( t) K _ {i} ( x t ) d t ,\ \ i = 1 \dots n , $$

where $ f _ {n+} 1 ( x) = f _ {1} ( x) $. Such a sequence of integral transforms is called a chain of integral transforms. For $ n = 2 $, repeated integral transforms are often called Fourier transforms.

Multiple (multi-dimensional) integral transforms are transforms (1) where $ t , x \in \mathbf R ^ {n} $ and $ C $ is some domain in the complex Euclidean $ n $- dimensional space.

Integral transforms of generalized functions can be constructed by the following basic methods:

1) One constructs a space of test functions $ U $ containing the kernel $ K ( x , t ) $ of the integral transform $ T $ under consideration. Then the transform $ T f $ for any generalized function $ f \in U ^ \prime $ is defined as the value of $ f $ on the test function $ K ( x , t ) $ according to the formula

$$ T [ f ( t) ] ( x) = \langle f , K ( x , t ) \rangle . $$

2) A space of test functions $ U $ is constructed on which the classical integral transform $ T $ is defined, mapping $ U $ onto some space of test functions $ V $. Then the integral transform $ T ^ \prime $ of a generalized function $ f \in V ^ \prime $ is defined by the equation

$$ \langle T ^ \prime f , \phi \rangle = \langle f , T \phi \rangle ,\ \ \phi \in U . $$

3) The required integral transform is expressed in terms of another integral transform that is defined for generalized functions.

See also Convolution transform; Euler transformation; Fourier transform; Gauss transform; Gegenbauer transform; Hardy transform; Hermite transform; Jacobi transform; Kontorovich–Lebedev transform; Mehler–Fock transform; Meijer transform; Mellin transform; Stieltjes transform; Watson transform; Whittaker transform.

References

[1] V.A. Ditkin, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–75 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82
[2] Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)
[3] V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)

Comments

References

[a1] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)
[a2] H. Zemanian, "Generalized integral transformations" , Interscience (1968)
How to Cite This Entry:
Integral transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_transform&oldid=47383
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article