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''Mehler–Fok transform, Fock–Mehler transform, Fok–Mehler transform''

The [[Integral transform|integral transform]]

\begin{equation*} F ( \tau ) = \frac { \pi } { 2 } \int _ { 0 } ^ { \infty } P _ { ( i \tau - 1 ) / 2 } ( 2 x ^ { 2 } + 1 ) f ( x ) d x, \end{equation*}

where $P _ { \nu } ( z )$ is the associated Legendre function of the first kind (cf. [[Legendre functions|Legendre functions]]). This transform was introduced by F.G. Mehler [[#References|[a1]]]. Some sufficient conditions for the inversion formula was found by V.A. Fock (also spelled V.A. Fok) [[#References|[a2]]] and N.N. Lebedev [[#References|[a3]]]. Some applications of the Mehler–Fock transform are given in [[#References|[a7]]].

If $f \in L _ { 2 } ( {\bf R} _ { + } ; x ^ { - 1 } )$, then the integral $F ( \tau )$ converges in the mean square with respect to the norm of the space $L _ { 2 } ( \mathbf{R} _ { + } ; \tau \operatorname { tanh } ( \pi \tau / 2 ) )$ and is an isomorphism between these spaces. Moreover, the [[Parseval equality|Parseval equality]] is true:

\begin{equation*} \int _ { 0 } ^ { \infty } | f ( x ) | ^ { 2 } \frac { d x } { x } = \frac { 4 } { \pi ^ { 2 } } \int _ { 0 } ^ { \infty } \tau \operatorname { tanh } ( \frac { \pi \tau } { 2 } ) \left| F ( \tau ) \right| ^ { 2 } d \tau, \end{equation*}

as well as the inversion formula

\begin{equation*} f ( x ) = \frac { 2 x } { \pi } \times \end{equation*}

\begin{equation*} \times \operatorname { lim } _ { N \rightarrow \infty } \int _ { 1 / N } ^ { N } \tau \operatorname { tanh } \left( \frac { \pi \tau } { 2 } \right) P _ { ( i \tau - 1 ) / 2 } ( 2 x ^ { 2 } + 1 ) F ( \tau ) d \tau , \end{equation*}

where the limit is taken with respect to the norm in $L _ { 2 } ( {\bf R} _ { + } ; x ^ { - 1 } )$. As is shown, for instance, in [[#References|[a5]]], the Mehler–Fock transform can be represented as the composition of the Hankel transform of index zero (cf. [[Integral transform|Integral transform]]; [[Hardy transform|Hardy transform]]) and the [[Kontorovich-Lebedev-transform(2)|Kontorovich–Lebedev transform]].

The generalized Mehler–Fock transform and its inverse involve the associated [[Legendre functions|Legendre functions]] of the first kind $P _ { \nu } ^ { ( k ) } ( x )$ and are accordingly defined as:

\begin{equation*} F ( \tau ) = \frac { \tau \operatorname { sinh } ( \pi \tau ) } { \pi } \Gamma \left( \frac { 1 } { 2 } - k + i \tau \right)\times \end{equation*}

\begin{equation*} \times\, \Gamma \left( \frac { 1 } { 2 } - k - i \tau \right) \int _ { 1 } ^ { \infty } P _ { i \tau - 1/2 } ^ { ( k ) } ( x ) f ( x ) d x ,\; f ( x ) = \int _ { 0 } ^ { \infty } P _ { i \tau -1/2} ^ { ( k ) }  ( x ) F ( \tau ) d \tau. \end{equation*}

If $k = 0$, these formulas reduce by simple substitutions to the ordinary Mehler–Fock transform. For $k = 1 / 2$, $x = \operatorname { cosh } \alpha$ one obtains the [[Fourier cosine transform|Fourier cosine transform]], while $k = - 1 / 2$, $x = \operatorname { cosh } \alpha$ leads to the [[Fourier sine transform|Fourier sine transform]].

If $f , g \in L _ { p } ( {\bf R} _ { + } ; x ^ { \nu p - 1 } )$, where $1 / 2 &lt; \nu &lt; 1$, $p \geq 1$, then for the Mehler–Fock transform of type (see [[#References|[a5]]])

\begin{equation*} F ( \tau ) = \int _ { 1 } ^ { \infty } P _ { i \tau - 1 / 2 } ( x ) f ( x ) d x \end{equation*}

one can define the convolution operator (cf. also [[Convolution transform|Convolution transform]])

\begin{equation*} ( f ^ { * } g ) ( x ) = \int _ { 1 } ^ { \infty } \int _ { 1 } ^ { \infty } S ( x , y , t ) f ( t ) g ( y ) d t d y, \end{equation*}

where $x &gt; 1$ and

\begin{equation*} S ( x , y , t ) = \sqrt { \frac { 2 \pi } { D } } \operatorname { log } \left( \frac { x + y + t + 1 + \sqrt { D } } { x + y + t + 1 - \sqrt { D } } \right), \end{equation*}

for $x , y , t \geq 1$ and $D = x ^ { 2 } + y ^ { 2 } + t ^ { 2 } - 1 - 2 x y t$, where the main values of the square and the logarithm are taken (cf. also [[Logarithmic function|Logarithmic function]]).

The convolution $( f ^ { * } g ) ( x )$ belongs to the space $L _ { p } ( \mathbf{R} _ { + } ; x ^ { ( 1 - \nu ) p - 1 } )$ and has the following representation:

\begin{equation*} ( f ^ { * } g ) ( x ) = \end{equation*}

\begin{equation*} = \pi ^ { 2 } \sqrt { \frac { \pi } { 2 } } \int _ { 0 } ^ { \infty } \tau \frac { \operatorname { sinh } ( \pi \tau ) } { \operatorname { cosh } ^ { 3 } ( \pi \tau ) } P _ { i \tau - 1 / 2 } ( x ) F ( \tau ) G ( \tau ) d \tau, \end{equation*}

where $G ( \tau )$ is the Mehler–Fock transform of the function $g$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  F.G. Mehler,   "Ueber eine mit den Kugel- und cylinderfunctionen verwandte Function und ihre Anwendung in der Theorie der Electricitätsvertheilung"  ''Math. Ann.'' , '''18'''  (1881)  pp. 161–194</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  V.A. Fock,   "On the representation of an arbitrary function by integrals involving the Legendre function with a complex index"  ''Dokl. Akad. Nauk SSSR'' , '''39''' :  7  (1943)  pp. 279–283  (In Russian)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  N.N. Lebedev,   "The Parseval theorem for the Mehler–Fock integral transform"  ''Dokl. Akad. Nauk SSSR'' , '''68'''  (1949)  pp. 445–448  (In Russian)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  S.B. Yakubovich,   "On the Mehler–Fock integral transform in $L _ { p }$-spaces"  ''Extracta Math.'' , '''8''' :  2–3  (1993)  pp. 162–164</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S.B. Yakubovich,   "Index transforms" , World Sci.  (1996)  pp. Chap. 3</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  F. Oberhettinger,   T.P. Higgins,   "Tables of Lebedev, Mehler and generalized Mehler transforms" , Boeing Sci. Res. Lab.  (1961)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  I.N. Sneddon,   "The use of integral transforms" , McGraw-Hill  (1972)  pp. Chap. 7</td></tr></table>