Integral transform

A transform of functions, having the form

(1)

where is a finite or infinite contour in the complex plane and is the kernel of the integral transform (cf. Kernel of an integral operator). In most cases one considers integral transforms for which and is the real axis or a part of it. If , then the transform is said to be finite. Formulas enabling one to recover the function from a known are called inversion formulas of the integral transform.

Examples of integral transforms. The Bochner transform:

where is the Bessel function of the first kind of order (cf. Bessel functions) and is the distance in . The inversion formula is: . The Parseval identity is:

The Weber transform:

where and and are the Bessel functions of first and second kind. The inversion formula is:

For , the Weber transform turns into the Hankel transform:

For this transform reduces to the Fourier sine and cosine transforms. The inversion formula is as follows: If , if is of bounded variation in a neighbourhood of a point and if , then

The Parseval identity: If , if and are the Hankel transforms of the functions and , where , then

Other forms of the Hankel transform are:

The Weierstrass transform:

it is a special case of a convolution transform.

Repeated transforms. Let

where . Such a sequence of integral transforms is called a chain of integral transforms. For , repeated integral transforms are often called Fourier transforms.

Multiple (multi-dimensional) integral transforms are transforms (1) where and is some domain in the complex Euclidean -dimensional space.

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

1) One constructs a space of test functions containing the kernel of the integral transform under consideration. Then the transform for any generalized function is defined as the value of on the test function according to the formula

2) A space of test functions is constructed on which the classical integral transform is defined, mapping onto some space of test functions . Then the integral transform of a generalized function is defined by the equation

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)

References

[a1]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)
[a2]	H. Zemanian, "Generalized integral transformations" , Interscience (1968)