Mehler-Fock-transform(2)

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

The integral transform

where 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 , then the integral converges in the mean square with respect to the norm of the space and is an isomorphism between these spaces. Moreover, the Parseval equality is true:

as well as the inversion formula

where the limit is taken with respect to the norm in . 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 and are accordingly defined as:

If , these formulas reduce by simple substitutions to the ordinary Mehler–Fock transform. For , one obtains the Fourier cosine transform, while , leads to the Fourier sine transform.

If , where , , then for the Mehler–Fock transform of type (see [a5])

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

where and

for and , where the main values of the square and the logarithm are taken (cf. also Logarithmic function).

The convolution belongs to the space and has the following representation:

where is the Mehler–Fock transform of the function .

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 -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