View source for Mehler-Fock-transform(2)

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

The [[Integral transform|integral transform]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m1201901.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m1201902.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m1201903.png" />, then the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m1201904.png" /> converges in the mean square with respect to the norm of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m1201905.png" /> and is an isomorphism between these spaces. Moreover, the [[Parseval equality|Parseval equality]] is true:

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as well as the inversion formula

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where the limit is taken with respect to the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m1201909.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019010.png" /> and are accordingly defined as:

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019012.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019013.png" />, these formulas reduce by simple substitutions to the ordinary Mehler–Fock transform. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019015.png" /> one obtains the [[Fourier cosine transform|Fourier cosine transform]], while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019017.png" /> leads to the [[Fourier sine transform|Fourier sine transform]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019020.png" />, then for the Mehler–Fock transform of type (see [[#References|[a5]]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019021.png" /></td> </tr></table>

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

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019023.png" /> and

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for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019026.png" />, where the main values of the square and the logarithm are taken (cf. also [[Logarithmic function|Logarithmic function]]).

The convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019027.png" /> belongs to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019028.png" /> and has the following representation:

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019030.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019031.png" /> is the Mehler–Fock transform of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019032.png" />.

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120190/m12019033.png" />-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>