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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009039.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009040.png" />, then at least one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009042.png" /> is equal to zero almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009043.png" /> (an analogue of the Titchmarsh theorem).
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009039.png" /> in the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009040.png" />, then at least one of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009042.png" /> is equal to zero almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009043.png" /> (an analogue of the Titchmarsh theorem).
  
The Kontorovich–Lebedev transform is the simplest and most basic in the class of integral transforms of non-convolution type, which forms a special class of so-called index transforms (cf. also [[Imbedding theorems for Orlicz–Sobolev spaces|Index transform]]), depending upon parameters, subscripts (indices) of the hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]) as kernels.
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The Kontorovich–Lebedev transform is the simplest and most basic in the class of integral transforms of non-convolution type, which forms a special class of so-called index transforms (cf. also [[Index transform|Index transform]]), depending upon parameters, subscripts (indices) of the hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]) as kernels.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.I. Kontorovich,  N.N. Lebedev,  "A method for the solution of problems in diffraction theory and related topics"  ''Zh. Eksper. Teor. Fiz.'' , '''8''' :  10–11  (1938)  pp. 1192–1206  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Lebedev,  "Sur une formule d'inversion"  ''Dokl. Akad. Sci. USSR'' , '''52'''  (1946)  pp. 655–658</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.N. Lebedev,  "Analog of the Parseval theorem for the one integral transform"  ''Dokl. Akad. Nauk SSSR'' , '''68''' :  4  (1949)  pp. 653–656  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Erdélyi,  W. Magnus,  F. Oberhettinger,  "Tables of integral transforms 1-2" , McGraw-Hill  (1954)  pp. Chap. XII</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chap. 6</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Vu Kim Tuan,  S.B. Yakubovich,  "The Kontorovich–Lebedev transform in a new class of functions"  ''Amer. Math. Soc. Transl.'' , '''137'''  (1987)  pp. 61–65</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.B. Yakubovich,  B. Fisher,  "On the Kontorovich–Lebedev transformation on distributions"  ''Proc. Amer. Math. Soc.'' , '''122''' :  3  (1994)  pp. 773–777</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.H. Zemanian,  "The Kontorovich–Lebedev transformation on distributions of compact support and its inversion"  ''Math. Proc. Cambridge Philos. Soc.'' , '''77'''  (1975)  pp. 139–143</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  S.B. Yakubovich,  Yu.F. Luchko,  "The hypergeometric approach to integral transforms and convolutions" , Kluwer Acad. Publ.  (1994)  pp. Chap. 6</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S.B. Yakubovich,  "Index transforms" , World Sci.  (1996)  pp. Chaps. 2;4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.I. Kontorovich,  N.N. Lebedev,  "A method for the solution of problems in diffraction theory and related topics"  ''Zh. Eksper. Teor. Fiz.'' , '''8''' :  10–11  (1938)  pp. 1192–1206  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.N. Lebedev,  "Sur une formule d'inversion"  ''Dokl. Akad. Sci. USSR'' , '''52'''  (1946)  pp. 655–658</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.N. Lebedev,  "Analog of the Parseval theorem for the one integral transform"  ''Dokl. Akad. Nauk SSSR'' , '''68''' :  4  (1949)  pp. 653–656  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Erdélyi,  W. Magnus,  F. Oberhettinger,  "Tables of integral transforms 1-2" , McGraw-Hill  (1954)  pp. Chap. XII</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chap. 6</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Vu Kim Tuan,  S.B. Yakubovich,  "The Kontorovich–Lebedev transform in a new class of functions"  ''Amer. Math. Soc. Transl.'' , '''137'''  (1987)  pp. 61–65</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S.B. Yakubovich,  B. Fisher,  "On the Kontorovich–Lebedev transformation on distributions"  ''Proc. Amer. Math. Soc.'' , '''122''' :  3  (1994)  pp. 773–777</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.H. Zemanian,  "The Kontorovich–Lebedev transformation on distributions of compact support and its inversion"  ''Math. Proc. Cambridge Philos. Soc.'' , '''77'''  (1975)  pp. 139–143</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  S.B. Yakubovich,  Yu.F. Luchko,  "The hypergeometric approach to integral transforms and convolutions" , Kluwer Acad. Publ.  (1994)  pp. Chap. 6</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S.B. Yakubovich,  "Index transforms" , World Sci.  (1996)  pp. Chaps. 2;4</TD></TR></table>

Latest revision as of 13:44, 14 November 2012

Lebedev–Kontorovich transform

The integral transform

where is the Macdonald function.

This transform was introduced in [a1] and later investigated in [a2]. If is an integrable function with the weight , i.e. , then is a bounded continuous function, which tends to zero at infinity (an analogue of the Riemann–Lebesgue lemma, cf. Fourier series, for the Fourier integral). If is a function of bounded variation in a neighbourhood of a point and if

then the following inversion formula holds:

If the Mellin transform of , denoted by , belongs to the space , then can be represented by an integral (see [a6]):

where is the Euler gamma-function.

Let . Then the integral converges in mean square and isomorphically maps the space onto the space . The inverse operator has the form [a10]

and the Parseval equality holds (see also [a3], [a5]):

The Kontorovich–Lebedev transform of distributions was considered in [a7], [a8]. A transform table for the Kontorovich–Lebedev transform can be found in [a4]. Special properties in -spaces are given in [a10].

For two functions , , define the operator of convolution for the Kontorovich–Lebedev transform as ([a9], [a10])

The following norm estimate is true:

and the space forms a normed ring with the convolution as operation of multiplication.

If , are the Kontorovich–Lebedev transforms of two functions , , then the factorization property is true:

If in the ring , then at least one of the functions , is equal to zero almost-everywhere on (an analogue of the Titchmarsh theorem).

The Kontorovich–Lebedev transform is the simplest and most basic in the class of integral transforms of non-convolution type, which forms a special class of so-called index transforms (cf. also Index transform), depending upon parameters, subscripts (indices) of the hypergeometric functions (cf. Hypergeometric function) as kernels.

References

[a1] M.I. Kontorovich, N.N. Lebedev, "A method for the solution of problems in diffraction theory and related topics" Zh. Eksper. Teor. Fiz. , 8 : 10–11 (1938) pp. 1192–1206 (In Russian)
[a2] N.N. Lebedev, "Sur une formule d'inversion" Dokl. Akad. Sci. USSR , 52 (1946) pp. 655–658
[a3] N.N. Lebedev, "Analog of the Parseval theorem for the one integral transform" Dokl. Akad. Nauk SSSR , 68 : 4 (1949) pp. 653–656 (In Russian)
[a4] A. Erdélyi, W. Magnus, F. Oberhettinger, "Tables of integral transforms 1-2" , McGraw-Hill (1954) pp. Chap. XII
[a5] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chap. 6
[a6] Vu Kim Tuan, S.B. Yakubovich, "The Kontorovich–Lebedev transform in a new class of functions" Amer. Math. Soc. Transl. , 137 (1987) pp. 61–65
[a7] S.B. Yakubovich, B. Fisher, "On the Kontorovich–Lebedev transformation on distributions" Proc. Amer. Math. Soc. , 122 : 3 (1994) pp. 773–777
[a8] A.H. Zemanian, "The Kontorovich–Lebedev transformation on distributions of compact support and its inversion" Math. Proc. Cambridge Philos. Soc. , 77 (1975) pp. 139–143
[a9] S.B. Yakubovich, Yu.F. Luchko, "The hypergeometric approach to integral transforms and convolutions" , Kluwer Acad. Publ. (1994) pp. Chap. 6
[a10] S.B. Yakubovich, "Index transforms" , World Sci. (1996) pp. Chaps. 2;4
How to Cite This Entry:
Kontorovich-Lebedev transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kontorovich-Lebedev_transform&oldid=22663
This article was adapted from an original article by S.B. Yakubovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article