View source for Olevskii transform

<!--This article has been texified automatically. Since there was no Nroff source code for this article, 
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
was used.
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.

Out of 17 formulas, 15 were replaced by TEX code.-->

{{TEX|semi-auto}}{{TEX|done}}
The [[Integral transform|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 [[#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]]].

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 \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,

\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====
<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>