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.

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)