# Integral transform

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.

#### 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)