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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|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|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|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|Convolution transform]]; [[Euler transformation|Euler transformation]]; [[Fourier transform|Fourier transform]]; [[Gauss transform|Gauss transform]]; [[Gegenbauer transform|Gegenbauer transform]]; [[Hardy transform|Hardy transform]]; [[Hermite transform|Hermite transform]]; [[Jacobi transform|Jacobi transform]]; [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev transform]]; [[Mehler–Fock transform|Mehler–Fock transform]]; [[Meijer transform|Meijer transform]]; [[Mellin transform|Mellin transform]]; [[Stieltjes transform|Stieltjes transform]]; [[Watson transform|Watson transform]]; [[Whittaker transform|Whittaker transform]].

====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>

====Comments====

====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>