Mehler-Fock transform

Mehler–Fok transform

The integral transform

$$ \tag{1 } F( x) = \int\limits _ { 0 } ^ \infty P _ {i \tau - 1/2 } ( x) f( \tau ) d \tau ,\ \ 1 \leq x < \infty , $$

where $ P _ \nu ( x) $ is the Legendre function of the first kind (cf. Legendre functions). If $ f \in L[ 0, \infty ) $, the function $ | f ^ { \prime } ( \tau ) | $ is locally integrable on $ [ 0, \infty ) $ and $ f( 0) = 0 $, then the following inversion formula is valid:

$$ \tag{2 } f( \tau ) = \tau \mathop{\rm tanh} \pi \tau \int\limits _ { 1 } ^ \infty P _ {i \tau - 1/2 } ( x) F( x) dx. $$

The Parseval identity. Consider the Mehler–Fock transform and its inverse defined by the equalities

$$ G( \tau ) = \int\limits _ { 1 } ^ \infty \sqrt {\tau \mathop{\rm tanh} \pi \tau } P _ {i \tau - 1/2 } ( x) g( x) dx, $$

$$ g( x) = \int\limits _ { 0 } ^ \infty \sqrt {\tau \mathop{\rm tanh} \ \pi \tau } P _ {i \tau - 1/2 } ( x) G( \tau ) d \tau . $$

If $ g _ {i} ( x) $, $ i = 1, 2 $, are arbitrary real-valued functions satisfying the conditions

$$ g _ {i} ( x) x ^ {-1/2} \mathop{\rm ln} ( 1+ x) \in L( 1, \infty ),\ \ g _ {i} ( x) \in L _ {2} ( 1, \infty ), $$

then

$$ \int\limits _ { 0 } ^ \infty G _ {1} ( \tau ) G _ {2} ( \tau ) d \tau = \ \int\limits _ { 1 } ^ \infty g _ {1} ( x) g _ {2} ( x) dx. $$

The generalized Mehler–Fock transform and the corresponding inversion formula are:

$$ \tag{3 } F( x) = \int\limits _ { 0 } ^ \infty P _ {i \tau - 1/2 } ^ {(k)} ( x) f( \tau ) d \tau , $$

and

$$ \tag{4 } f( \tau ) = \frac{1} \pi \tau \sinh \pi \tau \Gamma \left ( \frac{1}{2} - k + i \tau \right ) \Gamma \left ( \frac{1}{2} - k - i \tau \right ) \times $$

$$ \times \int\limits _ { 1 } ^ \infty P _ {i \tau - 1/2 } ^ {(k)} ( x) F( x) dx, $$

where $ P _ \nu ^ {(k)} ( x) $ are the associated Legendre functions of the first kind. For $ k= 0 $ formulas (3) and (4) reduce to (1) and (2); for $ k = 1/2 $, $ y = \cosh \alpha $, formulas (3) and (4) lead to the Fourier cosine transform, and for $ k = - 1/2 $, $ y = \cosh \alpha $ to the Fourier sine transform. The transforms (1) and (2) were introduced by F.G. Mehler [1]. The basic theorems were proved by V.A. Fock [V.A. Fok].

References

[1]	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
[2]	V.A. Fok, "On the representation of an arbitrary function by an integral involving Legendre functions with complex index" Dokl. Akad. Nauk SSSR , 39 (1943) pp. 253–256 (In Russian)
[3]	V.A. Ditkin, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–75 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82

References

[a1]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)