Mehler-Fock-transform(2)

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

The 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). This transform was introduced by F.G. Mehler [a1]. Some sufficient conditions for the inversion formula was found by V.A. Fock (also spelled V.A. Fok) [a2] and N.N. Lebedev [a3]. Some applications of the Mehler–Fock transform are given in [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 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 [a5], the Mehler–Fock transform can be represented as the composition of the Hankel transform of index zero (cf. Integral transform; Hardy transform) and the Kontorovich–Lebedev transform.

The generalized Mehler–Fock transform and its inverse involve the associated 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, while $k = - 1 / 2$, $x = \operatorname { cosh } \alpha$ leads to the Fourier sine transform.

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

\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 $x > 1$ 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).

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

[a1]	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
[a2]	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)
[a3]	N.N. Lebedev, "The Parseval theorem for the Mehler–Fock integral transform" Dokl. Akad. Nauk SSSR , 68 (1949) pp. 445–448 (In Russian)
[a4]	S.B. Yakubovich, "On the Mehler–Fock integral transform in $L _ { p }$-spaces" Extracta Math. , 8 : 2–3 (1993) pp. 162–164
[a5]	S.B. Yakubovich, "Index transforms" , World Sci. (1996) pp. Chap. 3
[a6]	F. Oberhettinger, T.P. Higgins, "Tables of Lebedev, Mehler and generalized Mehler transforms" , Boeing Sci. Res. Lab. (1961)
[a7]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chap. 7