View source for Kontorovich-Lebedev transform

''Lebedev–Kontorovich transform''

The [[Integral transform|integral transform]]

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200902.png" /> is the [[Macdonald function|Macdonald function]].

This transform was introduced in [[#References|[a1]]] and later investigated in [[#References|[a2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200903.png" /> is an integrable function with the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200904.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200905.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200906.png" /> is a bounded continuous function, which tends to zero at infinity (an analogue of the Riemann–Lebesgue lemma, cf. [[Fourier series|Fourier series]], for the [[Fourier integral|Fourier integral]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200907.png" /> is a [[Function of bounded variation|function of bounded variation]] in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k1200908.png" /> and if

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then the following inversion formula holds:

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If the [[Mellin transform|Mellin transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009013.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009014.png" />, belongs to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009016.png" /> can be represented by an integral (see [[#References|[a6]]]):

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009019.png" /> is the Euler [[Gamma-function|gamma-function]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009020.png" />. Then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009021.png" /> converges in mean square and isomorphically maps the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009022.png" /> onto the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009023.png" />. The inverse operator has the form [[#References|[a10]]]

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

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The Kontorovich–Lebedev transform of distributions was considered in [[#References|[a7]]], [[#References|[a8]]]. A transform table for the Kontorovich–Lebedev transform can be found in [[#References|[a4]]]. Special properties in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009026.png" />-spaces are given in [[#References|[a10]]].

For two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009028.png" />, define the operator of convolution for the Kontorovich–Lebedev transform as ([[#References|[a9]]], [[#References|[a10]]])

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The following norm estimate is true:

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and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009032.png" /> forms a [[Normed ring|normed ring]] with the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009033.png" /> as operation of multiplication.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009035.png" /> are the Kontorovich–Lebedev transforms of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k120/k120090/k12009037.png" />, then the factorization property is true:

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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).

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.

====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>