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The integral transform
 
The integral transform
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557801.png" /></td> </tr></table>
+
$$
 +
F ( \tau )  = \
 +
\int\limits _ { 0 } ^  \infty  K _ {i \tau }  ( x) f ( x)  d x ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557802.png" /> is the [[Macdonald function|Macdonald function]].
+
where $  K _  \nu  $
 +
is the [[Macdonald function|Macdonald function]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557803.png" /> is of bounded variation in a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557804.png" /> and if
+
If $  f $
 +
is of bounded variation in a neighbourhood of a point $  x = x _ {0} > 0 $
 +
and if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557805.png" /></td> </tr></table>
+
$$
 +
f ( x)  \mathop{\rm ln}  x  \in  L \left ( 0 ,
 +
\frac{1}{2}
 +
\right ) ,\ \
 +
f ( x) \sqrt x  \in  L \left (
 +
\frac{1}{2}
 +
, \infty \right ) ,
 +
$$
  
 
then the following inversion formula holds:
 
then the following inversion formula holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557806.png" /></td> </tr></table>
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557807.png" /></td> </tr></table>
+
\frac{f ( x _ {0} + ) + f ( x _ {0} - ) }{2 }
 +
=
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k0557809.png" />, be real-valued functions with
+
$$
 +
= \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578010.png" /></td> </tr></table>
+
\frac{2}{\pi  ^ {2} x _ {0} }
 +
\int\limits _ { 0 } ^  \infty  K _ {i \tau
 +
}  ( x _ {0} ) \tau  \sinh  \pi \tau F ( \tau )  d \tau .
 +
$$
 +
 
 +
Let  $  f _ {i} $,
 +
$  i = 1 , 2 $,
 +
be real-valued functions with
 +
 
 +
$$
 +
f _ {i} ( x) x  ^ {-} 3/4  \in  L ( 0 , \infty ) ,\ \
 +
f _ {i} ( x)  \in  L _ {2} ( 0 , \infty ) ;
 +
$$
  
 
and let
 
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578011.png" /></td> </tr></table>
+
$$
 +
F _ {i} ( \tau )  = \
 +
\int\limits _ { 0 } ^  \infty 
 +
 
 +
\frac{\sqrt {2 \tau  \sinh  \pi \tau } } \pi
 +
 
 +
\frac{K _ {i \tau }  }{\sqrt x }
 +
 
 +
f _ {i} ( x)  d x .
 +
$$
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578012.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
F _ {1} ( \tau ) F _ {2} ( \tau )  d \tau  = \
 +
\int\limits _ { 0 } ^  \infty 
 +
f _ {1} ( x ) f _ {2} ( x)  d x
 +
$$
  
 
(Parseval's identity).
 
(Parseval's identity).
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The finite Kontorovich–Lebedev transform has the form
 
The finite Kontorovich–Lebedev transform has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578013.png" /></td> </tr></table>
+
$$
 +
F ( \tau )  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578014.png" /></td> </tr></table>
+
\frac{2 \pi  \sinh  \pi \tau }{\pi  ^ {2} | I _ {i \alpha }  ( \alpha ) |  ^ {2} }
 +
\times
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055780/k05578016.png" /> is the modified Bessel function (see [[#References|[3]]]).
+
$$
 +
\times
 +
\int\limits _ { 0 } ^  \alpha 
 +
[ K _ {i \tau }  ( \alpha ) I _ {i \tau }  ( x) -
 +
I _ {i \tau }  ( \alpha ) K _ {i \tau }  ( x) ] f ( x) 
 +
\frac{dx}{x}
 +
,
 +
$$
 +
 
 +
$  \tau > 0 $,  
 +
where $  I _  \nu  $
 +
is the modified Bessel function (see [[#References|[3]]]).
  
 
The study of such transforms was initiated by M.I. Kontorovich and N.N. Lebedev (see [[#References|[1]]], [[#References|[2]]]).
 
The study of such transforms was initiated by M.I. Kontorovich and N.N. Lebedev (see [[#References|[1]]], [[#References|[2]]]).
Line 41: Line 109:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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. i. Toer. Fiz.'' , '''8''' :  10–11  (1938)  pp. 1192–1206  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Lebedev,  ''Dokl. Akad. Nauk SSSR'' , '''52''' :  5  (1945)  pp. 395–398</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Ya.S. Uflyand,  E. Yushkova,  ''Dokl. Akad. Nauk SSSR'' , '''164''' :  1  (1965)  pp. 70–72</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms and operational calculus" , Pergamon  (1965)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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. i. Toer. Fiz.'' , '''8''' :  10–11  (1938)  pp. 1192–1206  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Lebedev,  ''Dokl. Akad. Nauk SSSR'' , '''52''' :  5  (1945)  pp. 395–398</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Ya.S. Uflyand,  E. Yushkova,  ''Dokl. Akad. Nauk SSSR'' , '''164''' :  1  (1965)  pp. 70–72</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms and operational calculus" , Pergamon  (1965)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 22:15, 5 June 2020


The integral transform

$$ F ( \tau ) = \ \int\limits _ { 0 } ^ \infty K _ {i \tau } ( x) f ( x) d x , $$

where $ K _ \nu $ is the Macdonald function.

If $ f $ is of bounded variation in a neighbourhood of a point $ x = x _ {0} > 0 $ and if

$$ f ( x) \mathop{\rm ln} x \in L \left ( 0 , \frac{1}{2} \right ) ,\ \ f ( x) \sqrt x \in L \left ( \frac{1}{2} , \infty \right ) , $$

then the following inversion formula holds:

$$ \frac{f ( x _ {0} + ) + f ( x _ {0} - ) }{2 } = $$

$$ = \ \frac{2}{\pi ^ {2} x _ {0} } \int\limits _ { 0 } ^ \infty K _ {i \tau } ( x _ {0} ) \tau \sinh \pi \tau F ( \tau ) d \tau . $$

Let $ f _ {i} $, $ i = 1 , 2 $, be real-valued functions with

$$ f _ {i} ( x) x ^ {-} 3/4 \in L ( 0 , \infty ) ,\ \ f _ {i} ( x) \in L _ {2} ( 0 , \infty ) ; $$

and let

$$ F _ {i} ( \tau ) = \ \int\limits _ { 0 } ^ \infty \frac{\sqrt {2 \tau \sinh \pi \tau } } \pi \frac{K _ {i \tau } }{\sqrt x } f _ {i} ( x) d x . $$

Then

$$ \int\limits _ { 0 } ^ \infty F _ {1} ( \tau ) F _ {2} ( \tau ) d \tau = \ \int\limits _ { 0 } ^ \infty f _ {1} ( x ) f _ {2} ( x) d x $$

(Parseval's identity).

The finite Kontorovich–Lebedev transform has the form

$$ F ( \tau ) = \ \frac{2 \pi \sinh \pi \tau }{\pi ^ {2} | I _ {i \alpha } ( \alpha ) | ^ {2} } \times $$

$$ \times \int\limits _ { 0 } ^ \alpha [ K _ {i \tau } ( \alpha ) I _ {i \tau } ( x) - I _ {i \tau } ( \alpha ) K _ {i \tau } ( x) ] f ( x) \frac{dx}{x} , $$

$ \tau > 0 $, where $ I _ \nu $ is the modified Bessel function (see [3]).

The study of such transforms was initiated by M.I. Kontorovich and N.N. Lebedev (see [1], [2]).

References

[1] M.I. Kontorovich, N.N. Lebedev, "A method for the solution of problems in diffraction theory and related topics" Zh. Eksper. i. Toer. Fiz. , 8 : 10–11 (1938) pp. 1192–1206 (In Russian)
[2] N.N. Lebedev, Dokl. Akad. Nauk SSSR , 52 : 5 (1945) pp. 395–398
[3] Ya.S. Uflyand, E. Yushkova, Dokl. Akad. Nauk SSSR , 164 : 1 (1965) pp. 70–72
[4] V.A. Ditkin, A.P. Prudnikov, "Integral transforms and operational calculus" , Pergamon (1965) (Translated from Russian)

Comments

A transform table for the Kontorovich–Lebedev transform can be found in [a1]. A treatment in some detail of the transform is in [a2].

References

[a1] A. Erdelyi, W. Magnus, F. Oberhettinger, "Tables of integral transforms" , 1–2 , McGraw-Hill (1954) pp. Chapt. XII
[a2] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6
How to Cite This Entry:
Kontorovich-Lebedev-transform(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kontorovich-Lebedev-transform(2)&oldid=12838
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