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''Mehler–Fok transform''
 
''Mehler–Fok transform''
  
 
The [[Integral transform|integral 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/m063/m063340/m0633401.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F( x)  = \int\limits _ { 0 } ^  \infty  P _ {i \tau - 1/2 }  ( x) f( \tau )  d \tau ,\ \
 +
1 \leq  x < \infty ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m0633402.png" /> is the Legendre function of the first kind (cf. [[Legendre functions|Legendre functions]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m0633403.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m0633404.png" /> is locally integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m0633405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m0633406.png" />, then the following inversion formula is valid:
+
where $  P _  \nu  ( x) $
 +
is the Legendre function of the first kind (cf. [[Legendre functions|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:
  
<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/m063/m063340/m0633407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
f( \tau )  = \tau  \mathop{\rm tanh}  \pi \tau \int\limits _ { 1 } ^  \infty  P _ {i \tau - 1/2 }  ( x) F( x) dx.
 +
$$
  
 
The Parseval identity. Consider the Mehler–Fock transform and its inverse defined by the equalities
 
The Parseval identity. Consider the Mehler–Fock transform and its inverse defined by the equalities
  
<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/m063/m063340/m0633408.png" /></td> </tr></table>
+
$$
 +
G( \tau )  = \int\limits _ { 1 } ^  \infty  \sqrt {\tau  \mathop{\rm tanh}  \pi \tau } P _ {i \tau - 1/2 }
 +
( x) g( x)  dx,
 +
$$
  
<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/m063/m063340/m0633409.png" /></td> </tr></table>
+
$$
 +
g( x)  = \int\limits _ { 0 } ^  \infty  \sqrt {\tau  \mathop{\rm tanh} \
 +
\pi \tau } P _ {i \tau - 1/2 }  ( x) G( \tau )  d \tau .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334011.png" />, are arbitrary real-valued functions satisfying the conditions
+
If $  g _ {i} ( x) $,
 +
$  i = 1, 2 $,  
 +
are arbitrary real-valued functions satisfying the conditions
  
<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/m063/m063340/m06334012.png" /></td> </tr></table>
+
$$
 +
g _ {i} ( x) x  ^ {-} 1/2  \mathop{\rm ln} ( 1+ x)  \in  L( 1, \infty ),\ \
 +
g _ {i} ( x)  \in  L _ {2} ( 1, \infty ),
 +
$$
  
 
then
 
then
  
<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/m063/m063340/m06334013.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty  G _ {1} ( \tau ) G _ {2} ( \tau )  d \tau  = \
 +
\int\limits _ { 1 } ^  \infty  g _ {1} ( x) g _ {2} ( x)  dx.
 +
$$
  
 
The generalized Mehler–Fock transform and the corresponding inversion formula are:
 
The generalized Mehler–Fock transform and the corresponding inversion formula are:
  
<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/m063/m063340/m06334014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
F( x)  = \int\limits _ { 0 } ^  \infty  P _ {i \tau - 1/2 }  ^ {(} k) ( x) f( \tau ) d \tau ,
 +
$$
  
 
and
 
and
  
<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/m063/m063340/m06334015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
f( \tau )  =
 +
\frac{1} \pi
 +
\tau  \sinh  \pi \tau \Gamma \left (
 +
\frac{1}{2}
 +
- k + i \tau
 +
\right ) \Gamma \left (
 +
\frac{1}{2}
 +
- k - i \tau \right ) \times
 +
$$
  
<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/m063/m063340/m06334016.png" /></td> </tr></table>
+
$$
 +
\times
 +
\int\limits _ { 1 } ^  \infty  P _ {i \tau - 1/2 }  ^ {(} k) ( x) F( x)  dx,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334017.png" /> are the associated Legendre functions of the first kind. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334018.png" /> formulas (3) and (4) reduce to (1) and (2); for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334020.png" />, formulas (3) and (4) lead to the [[Fourier cosine transform|Fourier cosine transform]], and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063340/m06334022.png" /> to the [[Fourier sine transform|Fourier sine transform]]. The transforms (1) and (2) were introduced by F.G. Mehler [[#References|[1]]]. The basic theorems were proved by V.A. Fock [V.A. Fok].
+
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|Fourier cosine transform]], and for $  k = - 1/2 $,  
 +
$  y = \cosh  \alpha $
 +
to the [[Fourier sine transform|Fourier sine transform]]. The transforms (1) and (2) were introduced by F.G. Mehler [[#References|[1]]]. The basic theorems were proved by V.A. Fock [V.A. Fok].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.N. Sneddon,  "The use of integral transforms" , McGraw-Hill  (1972)</TD></TR></table>

Revision as of 08:00, 6 June 2020


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

Comments

References

[a1] I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972)
How to Cite This Entry:
Mehler-Fock transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mehler-Fock_transform&oldid=22805
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article