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Difference between revisions of "Integral transform"

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[ T f  ] ( r)  =  2 \pi r  ^ {1-} n/2
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| [ T f  ] ( r) |  ^ {2} r  ^ {k-1}  d r  = \  
 
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| f ( \rho ) |  ^ {2} \rho  ^ {k-} 1 d \rho .
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Latest revision as of 19:34, 17 January 2024


A transform of functions, having the form

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