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The integral transforms
 
The integral transforms
  
<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/j/j054/j054130/j0541301.png" /></td> </tr></table>
+
$$
 +
J \{ F ( x) \}  = \
 +
f ^ { ( \alpha , \beta ) } ( n)  = \
 +
\int\limits _ { - } 1 ^ { 1 }
 +
P _ {n} ^ {( \alpha , \beta ) } ( x) F ( x)  dx,
 +
$$
  
<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/j/j054/j054130/j0541302.png" /></td> </tr></table>
+
$$
 +
= 0, 1 \dots
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j0541303.png" /> are the [[Jacobi polynomials|Jacobi polynomials]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j0541304.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j0541305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j0541306.png" /> are real numbers. The inversion formula has the form
+
where the $  P _ {n} ^ {( \alpha , \beta ) } ( x) $
 +
are the [[Jacobi polynomials|Jacobi polynomials]] of degree $  n $,  
 +
and $  \alpha > - 1 $
 +
and $  \beta > - 1 $
 +
are real numbers. The inversion formula has the form
  
<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/j/j054/j054130/j0541307.png" /></td> </tr></table>
+
$$
 +
F ( x)  = \
 +
\sum _ {n = 0 } ^  \infty 
  
<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/j/j054/j054130/j0541308.png" /></td> </tr></table>
+
\frac{1}{\delta _ {n} }
  
<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/j/j054/j054130/j0541309.png" /></td> </tr></table>
+
( 1 - x)  ^  \alpha
 +
( 1 + x)  ^  \beta
 +
P _ {n} ^ {( \alpha , \beta ) } ( x)
 +
f ^ { ( \alpha , \beta ) } ( n),
 +
$$
 +
 
 +
$$
 +
- 1  < < 1,
 +
$$
 +
 
 +
$$
 +
\delta _ {n}  =
 +
\frac{2 ^ {\alpha + \beta + 1 } \Gamma
 +
( \alpha + n + 1) \Gamma ( \beta + n + 1) }{n! ( \alpha
 +
+ \beta + 2n + 1) \Gamma ( \alpha + \beta + n + 1) }
 +
,
 +
$$
  
 
provided the series converges.
 
provided the series converges.
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The Jacobi transform reduces the operation
 
The Jacobi transform reduces the operation
  
<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/j/j054/j054130/j05413010.png" /></td> </tr></table>
+
$$
 +
T [ F ( x)]  = \
 +
{
 +
\frac{d}{dx}
 +
}
 +
\left \{
 +
( 1 - x  ^ {2} )
 +
 
 +
\frac{dF }{dx }
 +
+
 +
[( \alpha - \beta ) + ( \alpha + \beta ) x]
 +
 
 +
\frac{dF }{dx }
 +
 
 +
\right \}
 +
$$
  
 
to an algebraic one by the formula
 
to an algebraic one by the formula
  
<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/j/j054/j054130/j05413011.png" /></td> </tr></table>
+
$$
 +
J \{ T [ F ( x)] \}  = -
 +
( n + 1) ( n + \alpha + \beta )
 +
f ^ { ( \alpha , \beta ) } ( n) +
 +
$$
  
<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/j/j054/j054130/j05413012.png" /></td> </tr></table>
+
$$
 +
+
 +
\left . \{ [( \alpha - \beta ) + ( \alpha +
 +
\beta ) x] P _ {n} ^ {( \alpha , \beta ) } ( x) F ( x) \} \right | _ {-} 1  ^ {1} .
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j05413013.png" /> the Jacobi transform is the [[Legendre transform|Legendre transform]]; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j05413014.png" /> it is the [[Gegenbauer transform|Gegenbauer transform]]. Jacobi transforms are used in solving differential equations containing the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054130/j05413015.png" />. The Jacobi transform has also been defined for a special class of generalized functions.
+
When $  \alpha = \beta = 0 $
 +
the Jacobi transform is the [[Legendre transform|Legendre transform]]; for $  \alpha = \beta = \nu - 1/2 $
 +
it is the [[Gegenbauer transform|Gegenbauer transform]]. Jacobi transforms are used in solving differential equations containing the operator $  T $.  
 +
The Jacobi transform has also been defined for a special class of generalized functions.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.J. Scott,  "Jacobi transforms"  ''Quart. J. Math.'' , '''4''' :  13  (1953)  pp. 36–40</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prundnikov,  "Integral transforms"  ''Progress in Math.''  (1969)  pp. 1–85  ''Itogi Nauk. Mat. Anal. 1966''  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transformations" , Interscience  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.J. Scott,  "Jacobi transforms"  ''Quart. J. Math.'' , '''4''' :  13  (1953)  pp. 36–40</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prundnikov,  "Integral transforms"  ''Progress in Math.''  (1969)  pp. 1–85  ''Itogi Nauk. Mat. Anal. 1966''  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transformations" , Interscience  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
See (the editorial comments to) [[Gegenbauer transform|Gegenbauer transform]]. Usually the Jacobi transform is written as
 
See (the editorial comments to) [[Gegenbauer transform|Gegenbauer transform]]. Usually the Jacobi transform is written as
  
<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/j/j054/j054130/j05413016.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { - } 1 ^ { 1 }  F ( x)
 +
P _ {n} ^ {( \alpha , \beta ) } ( x) ( 1 - x )  ^  \alpha
 +
( 1 + x )  ^  \beta  d x ,
 +
$$
  
 
which generalizes the expression given in [[Gegenbauer transform|Gegenbauer transform]].
 
which generalizes the expression given in [[Gegenbauer transform|Gegenbauer transform]].

Revision as of 22:14, 5 June 2020


The integral transforms

$$ J \{ F ( x) \} = \ f ^ { ( \alpha , \beta ) } ( n) = \ \int\limits _ { - } 1 ^ { 1 } P _ {n} ^ {( \alpha , \beta ) } ( x) F ( x) dx, $$

$$ n = 0, 1 \dots $$

where the $ P _ {n} ^ {( \alpha , \beta ) } ( x) $ are the Jacobi polynomials of degree $ n $, and $ \alpha > - 1 $ and $ \beta > - 1 $ are real numbers. The inversion formula has the form

$$ F ( x) = \ \sum _ {n = 0 } ^ \infty \frac{1}{\delta _ {n} } ( 1 - x) ^ \alpha ( 1 + x) ^ \beta P _ {n} ^ {( \alpha , \beta ) } ( x) f ^ { ( \alpha , \beta ) } ( n), $$

$$ - 1 < x < 1, $$

$$ \delta _ {n} = \frac{2 ^ {\alpha + \beta + 1 } \Gamma ( \alpha + n + 1) \Gamma ( \beta + n + 1) }{n! ( \alpha + \beta + 2n + 1) \Gamma ( \alpha + \beta + n + 1) } , $$

provided the series converges.

The Jacobi transform reduces the operation

$$ T [ F ( x)] = \ { \frac{d}{dx} } \left \{ ( 1 - x ^ {2} ) \frac{dF }{dx } + [( \alpha - \beta ) + ( \alpha + \beta ) x] \frac{dF }{dx } \right \} $$

to an algebraic one by the formula

$$ J \{ T [ F ( x)] \} = - ( n + 1) ( n + \alpha + \beta ) f ^ { ( \alpha , \beta ) } ( n) + $$

$$ + \left . \{ [( \alpha - \beta ) + ( \alpha + \beta ) x] P _ {n} ^ {( \alpha , \beta ) } ( x) F ( x) \} \right | _ {-} 1 ^ {1} . $$

When $ \alpha = \beta = 0 $ the Jacobi transform is the Legendre transform; for $ \alpha = \beta = \nu - 1/2 $ it is the Gegenbauer transform. Jacobi transforms are used in solving differential equations containing the operator $ T $. The Jacobi transform has also been defined for a special class of generalized functions.

References

[1] E.J. Scott, "Jacobi transforms" Quart. J. Math. , 4 : 13 (1953) pp. 36–40
[2] V.A. Ditkin, A.P. Prundnikov, "Integral transforms" Progress in Math. (1969) pp. 1–85 Itogi Nauk. Mat. Anal. 1966 (1967)
[3] A.G. Zemanian, "Generalized integral transformations" , Interscience (1968)

Comments

See (the editorial comments to) Gegenbauer transform. Usually the Jacobi transform is written as

$$ \int\limits _ { - } 1 ^ { 1 } F ( x) P _ {n} ^ {( \alpha , \beta ) } ( x) ( 1 - x ) ^ \alpha ( 1 + x ) ^ \beta d x , $$

which generalizes the expression given in Gegenbauer transform.

How to Cite This Entry:
Jacobi transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_transform&oldid=13591
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