Mehler-Fock-transform(2)
Mehler–Fok transform, Fock–Mehler transform, Fok–Mehler 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 ![]() |
[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 |
Mehler-Fock-transform(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mehler-Fock-transform(2)&oldid=50236