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Difference between revisions of "Olevskii transform"

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Out of 17 formulas, 15 were replaced by TEX code.-->
 
Out of 17 formulas, 15 were replaced by TEX code.-->
  
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The [[Integral transform|integral transform]]
 
The [[Integral transform|integral transform]]
  
\begin{equation} \tag{a1} F ( \tau ) = \end{equation}
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\begin{equation} \label{eq:a1} F ( \tau ) =  
 
 
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o120/o120020/o1200202.png"/></td> </tr></table>
 
  
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\frac{|\Gamma(c-a+\frac{i\tau}{2})|^2}{\Gamma(c)} \int_0^{\infty} x^{-a} (1+x)^{2a-c} {}_{2} F_{1} (a+\frac{i\tau}{2}, a-\frac{i\tau}{2} ; c ; -\frac{1}{x} )f(x) dx,
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\end{equation}
 
where ${}_{2} F_{1} (a, b ; c ; z )$ is a Gauss [[hypergeometric function]]. It was introduced by M.N. Olevskii in [[#References|[a1]]].
 
where ${}_{2} F_{1} (a, b ; c ; z )$ is a Gauss [[hypergeometric function]]. It was introduced by M.N. Olevskii in [[#References|[a1]]].
  
Letting $a = 1 / 2$, one obtains the [[Mehler–Fock transform]]. By changing the variable $x = \operatorname { sinh } ^ { - 2 } t$ and the respective parameters of the Gauss function, one obtains the Fourier–Jacobi transform [[#References|[a2]]].
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Letting $a = 1/2$, one obtains the [[Mehler–Fock transform]]. By changing the variable $x = \operatorname {sinh} ^ { - 2 } t$ and the respective parameters of the Gauss function, one obtains the Fourier–Jacobi transform [[#References|[a2]]].
  
 
One can show that the Olevskii transform is the composition of the [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev transform]] and the Hankel transform (cf. [[Integral transform]]; [[Hardy transform]]).
 
One can show that the Olevskii transform is the composition of the [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev transform]] and the Hankel transform (cf. [[Integral transform]]; [[Hardy transform]]).
  
The Gauss function in the integral (a1) is the hypergeometric series for $x > 1$ and for $0 < x \leq 1$ one can understand it as an [[analytic continuation]], which can be obtained from the Mellin–Barnes integral representation [[#References|[a3]]].
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The Gauss function in the integral \eqref{eq:a1} is the hypergeometric series for $x > 1$ and for $0 < x \leq 1$ one can understand it as an [[analytic continuation]], which can be obtained from the Mellin–Barnes integral representation [[#References|[a3]]].
  
 
The following integral transform is also called the Olevskii transform. It is an integral over the index $\tau$ of the Gauss function,
 
The following integral transform is also called the Olevskii transform. It is an integral over the index $\tau$ of the Gauss function,
  
\begin{equation} \tag{a2} F ( x ) = \frac { x ^ { - a } ( 1 + x ) ^ { 2 a - c } } { \Gamma ( c ) } \times \end{equation}
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\begin{equation} \label{eq:a2} F ( x ) = \frac { x ^ { - a } ( 1 + x ) ^ { 2 a - c } } { \Gamma ( c ) } \times  
  
\begin{equation*} \times \int _ { - \infty } ^ { \infty } \tau \left| \Gamma \left( c - a + \frac { i \tau } { 2 } \right) \right| ^ { 2 } \times \times \square _ { 2 } F _ { 1 } \left( a + \frac { i \tau } { 2 } , a - \frac { i \tau } { 2 } ; c ; - \frac { 1 } { x } \right) f ( \tau ) d \tau. \end{equation*}
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\int _ { - \infty } ^ { \infty } \tau \left| \Gamma \left( c - a + \frac{i \tau} {2} \right) \right|^{2} \times {}_{2} F_{1} \left(a + \frac {i \tau} {2} , a - \frac {i \tau} {2} ; c ; - \frac {1} {x} \right) f (\tau) d \tau. \end{equation}
  
 
Here, $f$ is an arbitrary odd function belonging to the space $L _ { 2 } ( \mathbf{R} ; \omega ( \tau ) )$, where
 
Here, $f$ is an arbitrary odd function belonging to the space $L _ { 2 } ( \mathbf{R} ; \omega ( \tau ) )$, where
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\begin{equation*} \omega ( \tau ) = \frac { \tau } { \operatorname { sinh } ( \pi \tau ) } \left| \frac { \Gamma ( c - a + \frac { i \tau } { 2 } ) } { \Gamma ( a + \frac { i \tau } { 2 } ) } \right| ^ { 2 } . \end{equation*}
 
\begin{equation*} \omega ( \tau ) = \frac { \tau } { \operatorname { sinh } ( \pi \tau ) } \left| \frac { \Gamma ( c - a + \frac { i \tau } { 2 } ) } { \Gamma ( a + \frac { i \tau } { 2 } ) } \right| ^ { 2 } . \end{equation*}
  
The transform (a2) maps this space onto the space $L _ { 2 } ( \mathbf{R} _ { + } ; x ^ { - 1 } ( 1 + x ) ^ { c - 2 a } )$ and the [[Parseval equality|Parseval equality]] holds:
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The transform \eqref{eq:a2} maps this space onto the space $L_{2} ( \mathbf{R} _ { + } ; x ^ { - 1 } ( 1 + x ) ^ { c - 2 a } )$ and the [[Parseval equality]] holds:
 
 
\begin{equation*} \int _ { 0 } ^ { \infty } | F ( x ) | ^ { 2 } ( 1 + x ) ^ { c - 2 a } \frac { d x } { x } = \end{equation*}
 
  
\begin{equation*} = 8 \pi ^ { 2 } \int _ { - \infty } ^ { \infty } \tau \operatorname { sinh } ( \pi \tau ) \left| \frac { \Gamma ( c - a + \frac { i \tau } { 2 } ) } { \Gamma ( a + \frac { i \tau } { 2 } ) } | ^ { 2 } \right| f ( \tau ) | ^ { 2 } d \tau. \end{equation*}
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\begin{equation*} \int _ { 0 } ^ { \infty } | F ( x ) | ^ { 2 } ( 1 + x ) ^ { c - 2 a } \frac { d x } { x } = 8 \pi ^ { 2 } \int _ { - \infty } ^ { \infty } \tau \operatorname { sinh } ( \pi \tau ) \left| \frac { \Gamma ( c - a + \frac { i \tau } { 2 } ) } { \Gamma ( a + \frac { i \tau } { 2 } ) } | ^ { 2 } \right| f ( \tau ) | ^ { 2 } d \tau. \end{equation*}
  
 
====References====
 
====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M.N. Olevskii,  "On the representation of an arbitrary function by integral with the kernel involving the hypergeometric function"  ''Dokl. Akad. Nauk SSSR'' , '''69''' :  1  (1949)  pp. 11–14  (In Russian)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  T.H. Koornwinder,  "Jacobi functions and analysis on noncompact semisimple Lie groups" , ''Special Functions: Group Theoretical Aspects and Applications'' , Reidel  (1984)  pp. 1–85</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S.B. Yakubovich,  "Index transforms" , World Sci.  (1996)  pp. Chap. 7</td></tr></table>
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<table>
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<tr><td valign="top">[a1]</td> <td valign="top">  M.N. Olevskii,  "On the representation of an arbitrary function by integral with the kernel involving the hypergeometric function"  ''Dokl. Akad. Nauk SSSR'' , '''69''' :  1  (1949)  pp. 11–14  (In Russian)</td></tr>
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<tr><td valign="top">[a2]</td> <td valign="top">  T.H. Koornwinder,  "Jacobi functions and analysis on noncompact semisimple Lie groups" , ''Special Functions: Group Theoretical Aspects and Applications'' , Reidel  (1984)  pp. 1–85</td></tr>
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<tr><td valign="top">[a3]</td> <td valign="top">  S.B. Yakubovich,  "Index transforms" , World Sci.  (1996)  pp. Chap. 7</td></tr>
 +
</table>

Latest revision as of 07:46, 25 November 2023

The integral transform

\begin{equation} \label{eq:a1} F ( \tau ) = \frac{|\Gamma(c-a+\frac{i\tau}{2})|^2}{\Gamma(c)} \int_0^{\infty} x^{-a} (1+x)^{2a-c} {}_{2} F_{1} (a+\frac{i\tau}{2}, a-\frac{i\tau}{2} ; c ; -\frac{1}{x} )f(x) dx, \end{equation} where ${}_{2} F_{1} (a, b ; c ; z )$ is a Gauss hypergeometric function. It was introduced by M.N. Olevskii in [a1].

Letting $a = 1/2$, one obtains the Mehler–Fock transform. By changing the variable $x = \operatorname {sinh} ^ { - 2 } t$ and the respective parameters of the Gauss function, one obtains the Fourier–Jacobi transform [a2].

One can show that the Olevskii transform is the composition of the Kontorovich–Lebedev transform and the Hankel transform (cf. Integral transform; Hardy transform).

The Gauss function in the integral \eqref{eq:a1} is the hypergeometric series for $x > 1$ and for $0 < x \leq 1$ one can understand it as an analytic continuation, which can be obtained from the Mellin–Barnes integral representation [a3].

The following integral transform is also called the Olevskii transform. It is an integral over the index $\tau$ of the Gauss function,

\begin{equation} \label{eq:a2} F ( x ) = \frac { x ^ { - a } ( 1 + x ) ^ { 2 a - c } } { \Gamma ( c ) } \times \int _ { - \infty } ^ { \infty } \tau \left| \Gamma \left( c - a + \frac{i \tau} {2} \right) \right|^{2} \times {}_{2} F_{1} \left(a + \frac {i \tau} {2} , a - \frac {i \tau} {2} ; c ; - \frac {1} {x} \right) f (\tau) d \tau. \end{equation}

Here, $f$ is an arbitrary odd function belonging to the space $L _ { 2 } ( \mathbf{R} ; \omega ( \tau ) )$, where

\begin{equation*} \omega ( \tau ) = \frac { \tau } { \operatorname { sinh } ( \pi \tau ) } \left| \frac { \Gamma ( c - a + \frac { i \tau } { 2 } ) } { \Gamma ( a + \frac { i \tau } { 2 } ) } \right| ^ { 2 } . \end{equation*}

The transform \eqref{eq:a2} maps this space onto the space $L_{2} ( \mathbf{R} _ { + } ; x ^ { - 1 } ( 1 + x ) ^ { c - 2 a } )$ and the Parseval equality holds:

\begin{equation*} \int _ { 0 } ^ { \infty } | F ( x ) | ^ { 2 } ( 1 + x ) ^ { c - 2 a } \frac { d x } { x } = 8 \pi ^ { 2 } \int _ { - \infty } ^ { \infty } \tau \operatorname { sinh } ( \pi \tau ) \left| \frac { \Gamma ( c - a + \frac { i \tau } { 2 } ) } { \Gamma ( a + \frac { i \tau } { 2 } ) } | ^ { 2 } \right| f ( \tau ) | ^ { 2 } d \tau. \end{equation*}

References

[a1] M.N. Olevskii, "On the representation of an arbitrary function by integral with the kernel involving the hypergeometric function" Dokl. Akad. Nauk SSSR , 69 : 1 (1949) pp. 11–14 (In Russian)
[a2] T.H. Koornwinder, "Jacobi functions and analysis on noncompact semisimple Lie groups" , Special Functions: Group Theoretical Aspects and Applications , Reidel (1984) pp. 1–85
[a3] S.B. Yakubovich, "Index transforms" , World Sci. (1996) pp. Chap. 7
How to Cite This Entry:
Olevskii transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Olevskii_transform&oldid=54667
This article was adapted from an original article by S.B. Yakubovich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article