Olevskii transform

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