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Difference between revisions of "Lebedev transform"

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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/l/l057/l057790/l0577901.png" /></td> </tr></table>
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$$F(\tau)=\int\limits_0^\infty[I_{i\tau}(x)+I_{-i\tau}(x)]K_{i\tau}(x)f(x)dx,\quad0\leq\tau<\infty,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057790/l0577902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057790/l0577903.png" /> are the modified [[Cylinder functions|cylinder functions]]. It was introduced by N.N. Lebedev [[#References|[1]]]. If
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where $I_\nu(x)$ and $K_\nu(x)$ are the modified [[Cylinder functions|cylinder functions]]. It was introduced by N.N. Lebedev [[#References|[1]]]. 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/l/l057/l057790/l0577904.png" /></td> </tr></table>
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$$x^{-1/2}f(x)\in L(0,1),\quad x^{1/2}f(x)\in L(1,\infty),$$
  
then for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057790/l0577905.png" /> one has the inversion formula
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then for almost-all $x$ one has the inversion formula
  
<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/l/l057/l057790/l0577906.png" /></td> </tr></table>
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$$f(x)=-\frac{4}{\pi^2}\int\limits_0^\infty F(\tau)\tau\sinh\pi\tau K_{i\tau}^2(x)d\tau.$$
  
 
====References====
 
====References====
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The following transform pair is also called a Lebedev transform (or [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev transform]])
 
The following transform pair is also called a Lebedev transform (or [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev 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/l/l057/l057790/l0577907.png" /></td> </tr></table>
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$$G(\tau)=\int\limits_0^\infty g(x)x^{-1/2}K_{i\tau}(x)dx,$$
  
<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/l/l057/l057790/l0577908.png" /></td> </tr></table>
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$$g(x)=\frac{2}{\pi^2}\frac{1}{\sqrt x}\int\limits_0^\infty\tau\sinh\pi\tau K_{i\tau}(x)G(\tau)d\tau.$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Lebedev,  "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chapt. 6</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.N. Lebedev,  "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chapt. 6</TD></TR></table>

Latest revision as of 15:43, 11 August 2014

The integral transform

$$F(\tau)=\int\limits_0^\infty[I_{i\tau}(x)+I_{-i\tau}(x)]K_{i\tau}(x)f(x)dx,\quad0\leq\tau<\infty,$$

where $I_\nu(x)$ and $K_\nu(x)$ are the modified cylinder functions. It was introduced by N.N. Lebedev [1]. If

$$x^{-1/2}f(x)\in L(0,1),\quad x^{1/2}f(x)\in L(1,\infty),$$

then for almost-all $x$ one has the inversion formula

$$f(x)=-\frac{4}{\pi^2}\int\limits_0^\infty F(\tau)\tau\sinh\pi\tau K_{i\tau}^2(x)d\tau.$$

References

[1] N.N. Lebedev, "On an integral representation of an arbitrary function in terms of squares of MacDonald functions with imaginary index" Sibirsk. Mat. Zh. , 3 : 2 (1962) pp. 213–222 (In Russian)


Comments

The following transform pair is also called a Lebedev transform (or Kontorovich–Lebedev transform)

$$G(\tau)=\int\limits_0^\infty g(x)x^{-1/2}K_{i\tau}(x)dx,$$

$$g(x)=\frac{2}{\pi^2}\frac{1}{\sqrt x}\int\limits_0^\infty\tau\sinh\pi\tau K_{i\tau}(x)G(\tau)d\tau.$$

References

[a1] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
[a2] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6
How to Cite This Entry:
Lebedev transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lebedev_transform&oldid=16220
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