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[[Orthogonal polynomials|Orthogonal polynomials]] on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j0541001.png" /> with the weight function
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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/j/j054/j054100/j0541002.png" /></td> </tr></table>
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 +
[[Orthogonal polynomials|Orthogonal polynomials]] on the interval  $  [- 1, 1] $
 +
with the weight function
 +
 
 +
$$
 +
h ( x)  = ( 1 - x)  ^  \alpha  ( 1 + x)  ^  \beta  ,\ \
 +
\alpha , \beta > - 1,\ \
 +
x \in [- 1, 1].
 +
$$
  
 
The standardized Jacobi polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]:
 
The standardized Jacobi polynomials are defined by the [[Rodrigues formula|Rodrigues 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/j054100/j0541003.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( x; \alpha , \beta )  = \
 +
P _ {n} ^ {( \alpha , \beta ) } ( x) =
 +
$$
  
<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/j054100/j0541004.png" /></td> </tr></table>
+
$$
 +
= \
 +
 
 +
\frac{(- 1)  ^ {n} }{n! 2  ^ {n} }
 +
( 1 - x) ^ {- \alpha
 +
} ( 1 + x) ^ {- \beta }
 +
\frac{d  ^ {n} }{dx  ^ {n} }
 +
[( 1 -
 +
x)  ^  \alpha  ( 1 + x )  ^  \beta  ( 1 - x  ^ {2} )  ^ {n} ],
 +
$$
  
 
and the orthonormal Jacobi polynomials have the form
 
and the orthonormal Jacobi polynomials have 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/j054100/j0541005.png" /></td> </tr></table>
+
$$
 +
\widehat{P}  _ {n} ( x; \alpha , \beta ) =
 +
$$
  
<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/j054100/j0541006.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sqrt {
 +
\frac{n! ( \alpha + \beta + 2n + 1) \Gamma ( \alpha
 +
+ \beta + n + 1) }{2 ^ {\alpha + \beta + 1 }
 +
\Gamma ( \alpha + n + 1) \Gamma ( \beta + n + 1) }
 +
} P _ {n} ( x; \alpha , \beta ).
 +
$$
  
The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j0541007.png" /> satisfies the differential equation
+
The polynomial $  P _ {n} ( x;  \alpha , \beta ) $
 +
satisfies the differential equation
  
<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/j054100/j0541008.png" /></td> </tr></table>
+
$$
 +
( 1 - x  ^ {2} ) y  ^ {\prime\prime} +
 +
[ \beta - \alpha - ( \alpha + \beta + 2) x]
 +
y  ^  \prime  + n ( n + \alpha + \beta + 1) y  = 0.
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j0541009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410010.png" />, the orthonormal Jacobi polynomials satisfy the following weighted estimate:
+
When $  \alpha \geq  - 1/2 $
 +
and $  \beta \geq  - 1/2 $,  
 +
the orthonormal Jacobi polynomials satisfy the following weighted estimate:
  
<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/j054100/j05410011.png" /></td> </tr></table>
+
$$
 +
( 1 - x) ^ {( 2 \alpha + 1)/4 }
 +
( 1 + x) ^ {( 2 \beta + 1)/4 }
 +
| \widehat{P}  _ {n} ( x;  \alpha , \beta ) |  \leq  c _ {1} ,
 +
$$
  
<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/j054100/j05410012.png" /></td> </tr></table>
+
$$
 +
x  \in  [- 1, 1],
 +
$$
  
where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410013.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410015.png" />. At <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410016.png" /> the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410017.png" /> grows at a rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410019.png" />, respectively.
+
where the constant $  c _ {1} $
 +
does not depend on $  n $
 +
and $  x $.  
 +
At $  x = \pm  1 $
 +
the sequence $  \{ \widehat{P}  _ {n} ( x;  \alpha , \beta ) \} $
 +
grows at a rate $  n ^ {\alpha + 1/2 } $
 +
and $  n ^ {\beta + 1/2 } $,  
 +
respectively.
  
Fourier series in Jacobi polynomials (cf. [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]) inside the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410020.png" /> are similar to trigonometric Fourier series. But in neighbourhoods of the end points of this interval, the orthogonality properties of Fourier–Jacobi series are different, because at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410021.png" /> the orthonormal Jacobi polynomials grow unboundedly. The Fourier–Jacobi series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410022.png" /> is uniformly convergent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410024.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410025.png" /> times continuously differentiable on this segment and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410026.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410027.png" />, where
+
Fourier series in Jacobi polynomials (cf. [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]) inside the interval $  (- 1, 1) $
 +
are similar to trigonometric Fourier series. But in neighbourhoods of the end points of this interval, the orthogonality properties of Fourier–Jacobi series are different, because at $  x = \pm  1 $
 +
the orthonormal Jacobi polynomials grow unboundedly. The Fourier–Jacobi series of a function $  f $
 +
is uniformly convergent on $  [- 1, 1] $
 +
if $  f $
 +
is $  p $
 +
times continuously differentiable on this segment and $  f ^ { ( p) } \in  \mathop{\rm Lip}  \gamma $
 +
with  $  p + \gamma > q + 1/2 $,  
 +
where
  
<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/j054100/j05410028.png" /></td> </tr></table>
+
$$
 +
= \max \{ \alpha , \beta \}  > - {
 +
\frac{1}{2}
 +
} .
 +
$$
  
 
Under these conditions the following inequality holds:
 
Under these conditions the following inequality holds:
  
<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/j054100/j05410029.png" /></td> </tr></table>
+
$$
 +
\left | f ( x) -
 +
\sum _ {k = 0 } ^ { n }
 +
a _ {k} \widehat{P}  _ {k} ( x;  \alpha , \beta ) \right |  \leq  \
  
<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/j054100/j05410030.png" /></td> </tr></table>
+
\frac{c _ {2} }{n ^ {p + \gamma } }
 +
n ^ {( 2q + 1)/2 } ,
 +
$$
  
where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410031.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410033.png" />. On the other hand, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410035.png" />, the remainder in the Fourier–Jacobi series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410036.png" /> satisfies the following weighted estimate:
+
$$
 +
x  \in  [- 1, 1],
 +
$$
  
<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/j054100/j05410037.png" /></td> </tr></table>
+
where the constant  $  c _ {2} $
 +
does not depend on  $  n $
 +
and  $  x $.
 +
On the other hand, when  $  \alpha \geq  - 1/2 $
 +
and  $  \beta \geq  - 1/2 $,
 +
the remainder in the Fourier–Jacobi series for  $  f $
 +
satisfies the following weighted estimate:
  
<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/j054100/j05410038.png" /></td> </tr></table>
+
$$
 +
( 1 - x  ^ {2} )  ^ {1/4}
 +
\sqrt {h ( x) }  \left | f ( x) -
 +
\sum _ {k = 0 } ^ { n }
 +
a _ {k} \widehat{P}  _ {k} ( x;  \alpha , \beta ) \right | \leq
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410039.png" />, the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410040.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410043.png" /> is the best uniform approximation error (cf. [[Best approximation|Best approximation]]) of the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410044.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410045.png" /> by polynomials of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410046.png" />.
+
$$
 +
\leq  \
 +
c _ {3} E _ {n} ( f  )  \mathop{\rm ln}  n,\ \
 +
x \in [- 1, 1],
 +
$$
  
The Jacobi polynomials were introduced by C.G.J. Jacobi [[#References|[1]]] in connection with the solution of the [[Hypergeometric equation|hypergeometric equation]]. Special cases of the Jacobi polynomials are: the [[Legendre polynomials|Legendre polynomials]] (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410047.png" />); the [[Chebyshev polynomials|Chebyshev polynomials]] of the first kind (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410048.png" />); the Chebyshev polynomials of the second kind (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410049.png" />); and the [[Ultraspherical polynomials|ultraspherical polynomials]] (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410050.png" />).
+
where  $  n \geq  2 $,
 +
the constant  $  c _ {3} $
 +
does not depend on  $  n $
 +
and  $  x $,
 +
and  $  E _ {n} ( f  ) $
 +
is the best uniform approximation error (cf. [[Best approximation|Best approximation]]) of the continuous function  $  f $
 +
on  $  [- 1, 1] $
 +
by polynomials of degree not exceeding  $  n $.
 +
 
 +
The Jacobi polynomials were introduced by C.G.J. Jacobi [[#References|[1]]] in connection with the solution of the [[Hypergeometric equation|hypergeometric equation]]. Special cases of the Jacobi polynomials are: the [[Legendre polynomials|Legendre polynomials]] (when $  \alpha = \beta = 0 $);  
 +
the [[Chebyshev polynomials|Chebyshev polynomials]] of the first kind (when $  \alpha = \beta = - 1/2 $);  
 +
the Chebyshev polynomials of the second kind (when $  \alpha = \beta = 1/2 $);  
 +
and the [[Ultraspherical polynomials|ultraspherical polynomials]] (when $  \alpha = \beta $).
  
 
See also [[Classical orthogonal polynomials|Classical orthogonal polynomials]].
 
See also [[Classical orthogonal polynomials|Classical orthogonal polynomials]].
Line 52: Line 156:
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G.J. Jacobi,  "Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe"  ''J. Reine Angew. Math.'' , '''56'''  (1859)  pp. 149–165</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Suetin,  "Classical orthogonal polynomials" , Moscow  (1978)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G.J. Jacobi,  "Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe"  ''J. Reine Angew. Math.'' , '''56'''  (1859)  pp. 149–165</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Suetin,  "Classical orthogonal polynomials" , Moscow  (1978)  (In Russian)</TD></TR></table>
  
 +
====Comments====
 +
See also [[#References|[a4]]], [[#References|[a1]]] and [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]].
 +
 +
Let  $  \alpha , \beta > - 1 $
 +
and  $  - 1 < x, y < 1 $.
 +
Then there is a product formula of the form
  
 +
$$
  
====Comments====
+
\frac{P _ {n} ^ {( \alpha , \beta ) } ( x) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) }
See also [[#References|[a4]]], [[#References|[a1]]] and [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]].
 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410052.png" />. Then there is a product formula of the form
+
\frac{P _ {n} ^ {( \alpha , \beta ) } ( y) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) }
 +
  = \
 +
\int\limits _ { - } 1 ^ { 1 }
  
<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/j054100/j05410053.png" /></td> </tr></table>
+
\frac{P _ {n} ^ {( \alpha , \beta ) } ( z) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) }
 +
\
 +
d \mu _ {x,y} ( z),\ \
 +
n = 0, 1 \dots
 +
$$
  
with positive measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410054.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410055.png" /> and either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410056.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410057.png" />. This yields a positive convolution structure for Jacobi series. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410058.png" /> the above measure can be computed explicitly from the addition formula for Jacobi polynomials. See [[#References|[a1]]], Lecture 4.
+
with positive measure $  d \mu _ {x,y} ( z) = d \mu _ {x,y} ^ {\alpha , \beta } ( z) $
 +
if and only if $  \alpha \geq  \beta $
 +
and either $  \beta \geq  - 1/2 $
 +
or $  \alpha + \beta \geq  0 $.  
 +
This yields a positive convolution structure for Jacobi series. For $  \alpha \geq  \beta \geq  - 1/2 $
 +
the above measure can be computed explicitly from the addition formula for Jacobi polynomials. See [[#References|[a1]]], Lecture 4.
  
 
For the dual problem one has
 
For the dual problem one has
  
<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/j054100/j05410059.png" /></td> </tr></table>
+
$$
 +
P _ {n} ^ {( \alpha , \beta ) } ( x)
 +
P _ {m} ^ {( \alpha , \beta ) } ( x)  = \
 +
\sum _ {k = | n - m | } ^ { {n }  + m }
 +
C ( k, m, n)
 +
P _ {k} ^ {( \alpha , \beta ) } ( x) ,
 +
$$
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410060.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410062.png" />. This yields a positive dual convolution structure for Jacobi series. See [[#References|[a1]]], Lecture 5.
+
with $  C ( k, m, n) \geq  0 $
 +
if $  \alpha \geq  \beta > - 1 $,  
 +
$  \alpha + \beta \geq  - 1 $.  
 +
This yields a positive dual convolution structure for Jacobi series. See [[#References|[a1]]], Lecture 5.
  
Jacobi polynomials admit many different group-theoretic interpretations. The three most important ones are as matrix elements of the irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410063.png" /> (cf. [[#References|[a5]]], Chapt. 3), as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410064.png" />-invariant [[Spherical harmonics|spherical harmonics]] on the unit sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/j/j054/j054100/j05410065.png" /> (cf. [[#References|[a2]]]) and as zonal spherical functions on the compact symmetric spaces of rank one (cf. [[#References|[a3]]], Chapt. 5, §4.3).
+
Jacobi polynomials admit many different group-theoretic interpretations. The three most important ones are as matrix elements of the irreducible representations of $  \mathop{\rm SU} ( 2) $(
 +
cf. [[#References|[a5]]], Chapt. 3), as $  O ( p) \times O ( q) $-
 +
invariant [[Spherical harmonics|spherical harmonics]] on the unit sphere in $  \mathbf R ^ {p + 1 } $(
 +
cf. [[#References|[a2]]]) and as zonal spherical functions on the compact symmetric spaces of rank one (cf. [[#References|[a3]]], Chapt. 5, §4.3).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Askey,  "Orthogonal polynomials and special functions" , ''Reg. Conf. Ser. Appl. Math.'' , '''21''' , SIAM  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L.J. Braaksma,  B. Meulenbeld,  "Jacobi polynomials as spherical harmonics"  ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''71'''  (1968)  pp. 384–389</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Helgason,  "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N.Ya. Vilenkin,  "Special functions and the theory of group representations" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Askey,  "Orthogonal polynomials and special functions" , ''Reg. Conf. Ser. Appl. Math.'' , '''21''' , SIAM  (1975)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B.L.J. Braaksma,  B. Meulenbeld,  "Jacobi polynomials as spherical harmonics"  ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''71'''  (1968)  pp. 384–389</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Helgason,  "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Szegö,  "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N.Ya. Vilenkin,  "Special functions and the theory of group representations" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR></table>

Latest revision as of 22:14, 5 June 2020


Orthogonal polynomials on the interval $ [- 1, 1] $ with the weight function

$$ h ( x) = ( 1 - x) ^ \alpha ( 1 + x) ^ \beta ,\ \ \alpha , \beta > - 1,\ \ x \in [- 1, 1]. $$

The standardized Jacobi polynomials are defined by the Rodrigues formula:

$$ P _ {n} ( x; \alpha , \beta ) = \ P _ {n} ^ {( \alpha , \beta ) } ( x) = $$

$$ = \ \frac{(- 1) ^ {n} }{n! 2 ^ {n} } ( 1 - x) ^ {- \alpha } ( 1 + x) ^ {- \beta } \frac{d ^ {n} }{dx ^ {n} } [( 1 - x) ^ \alpha ( 1 + x ) ^ \beta ( 1 - x ^ {2} ) ^ {n} ], $$

and the orthonormal Jacobi polynomials have the form

$$ \widehat{P} _ {n} ( x; \alpha , \beta ) = $$

$$ = \ \sqrt { \frac{n! ( \alpha + \beta + 2n + 1) \Gamma ( \alpha + \beta + n + 1) }{2 ^ {\alpha + \beta + 1 } \Gamma ( \alpha + n + 1) \Gamma ( \beta + n + 1) } } P _ {n} ( x; \alpha , \beta ). $$

The polynomial $ P _ {n} ( x; \alpha , \beta ) $ satisfies the differential equation

$$ ( 1 - x ^ {2} ) y ^ {\prime\prime} + [ \beta - \alpha - ( \alpha + \beta + 2) x] y ^ \prime + n ( n + \alpha + \beta + 1) y = 0. $$

When $ \alpha \geq - 1/2 $ and $ \beta \geq - 1/2 $, the orthonormal Jacobi polynomials satisfy the following weighted estimate:

$$ ( 1 - x) ^ {( 2 \alpha + 1)/4 } ( 1 + x) ^ {( 2 \beta + 1)/4 } | \widehat{P} _ {n} ( x; \alpha , \beta ) | \leq c _ {1} , $$

$$ x \in [- 1, 1], $$

where the constant $ c _ {1} $ does not depend on $ n $ and $ x $. At $ x = \pm 1 $ the sequence $ \{ \widehat{P} _ {n} ( x; \alpha , \beta ) \} $ grows at a rate $ n ^ {\alpha + 1/2 } $ and $ n ^ {\beta + 1/2 } $, respectively.

Fourier series in Jacobi polynomials (cf. Fourier series in orthogonal polynomials) inside the interval $ (- 1, 1) $ are similar to trigonometric Fourier series. But in neighbourhoods of the end points of this interval, the orthogonality properties of Fourier–Jacobi series are different, because at $ x = \pm 1 $ the orthonormal Jacobi polynomials grow unboundedly. The Fourier–Jacobi series of a function $ f $ is uniformly convergent on $ [- 1, 1] $ if $ f $ is $ p $ times continuously differentiable on this segment and $ f ^ { ( p) } \in \mathop{\rm Lip} \gamma $ with $ p + \gamma > q + 1/2 $, where

$$ q = \max \{ \alpha , \beta \} > - { \frac{1}{2} } . $$

Under these conditions the following inequality holds:

$$ \left | f ( x) - \sum _ {k = 0 } ^ { n } a _ {k} \widehat{P} _ {k} ( x; \alpha , \beta ) \right | \leq \ \frac{c _ {2} }{n ^ {p + \gamma } } n ^ {( 2q + 1)/2 } , $$

$$ x \in [- 1, 1], $$

where the constant $ c _ {2} $ does not depend on $ n $ and $ x $. On the other hand, when $ \alpha \geq - 1/2 $ and $ \beta \geq - 1/2 $, the remainder in the Fourier–Jacobi series for $ f $ satisfies the following weighted estimate:

$$ ( 1 - x ^ {2} ) ^ {1/4} \sqrt {h ( x) } \left | f ( x) - \sum _ {k = 0 } ^ { n } a _ {k} \widehat{P} _ {k} ( x; \alpha , \beta ) \right | \leq $$

$$ \leq \ c _ {3} E _ {n} ( f ) \mathop{\rm ln} n,\ \ x \in [- 1, 1], $$

where $ n \geq 2 $, the constant $ c _ {3} $ does not depend on $ n $ and $ x $, and $ E _ {n} ( f ) $ is the best uniform approximation error (cf. Best approximation) of the continuous function $ f $ on $ [- 1, 1] $ by polynomials of degree not exceeding $ n $.

The Jacobi polynomials were introduced by C.G.J. Jacobi [1] in connection with the solution of the hypergeometric equation. Special cases of the Jacobi polynomials are: the Legendre polynomials (when $ \alpha = \beta = 0 $); the Chebyshev polynomials of the first kind (when $ \alpha = \beta = - 1/2 $); the Chebyshev polynomials of the second kind (when $ \alpha = \beta = 1/2 $); and the ultraspherical polynomials (when $ \alpha = \beta $).

See also Classical orthogonal polynomials.

References

[1] C.G.J. Jacobi, "Untersuchungen über die Differentialgleichung der hypergeometrischen Reihe" J. Reine Angew. Math. , 56 (1859) pp. 149–165
[2] P.K. Suetin, "Classical orthogonal polynomials" , Moscow (1978) (In Russian)

Comments

See also [a4], [a1] and Fourier series in orthogonal polynomials.

Let $ \alpha , \beta > - 1 $ and $ - 1 < x, y < 1 $. Then there is a product formula of the form

$$ \frac{P _ {n} ^ {( \alpha , \beta ) } ( x) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) } \frac{P _ {n} ^ {( \alpha , \beta ) } ( y) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) } = \ \int\limits _ { - } 1 ^ { 1 } \frac{P _ {n} ^ {( \alpha , \beta ) } ( z) }{P _ {n} ^ {( \alpha , \beta ) } ( 1) } \ d \mu _ {x,y} ( z),\ \ n = 0, 1 \dots $$

with positive measure $ d \mu _ {x,y} ( z) = d \mu _ {x,y} ^ {\alpha , \beta } ( z) $ if and only if $ \alpha \geq \beta $ and either $ \beta \geq - 1/2 $ or $ \alpha + \beta \geq 0 $. This yields a positive convolution structure for Jacobi series. For $ \alpha \geq \beta \geq - 1/2 $ the above measure can be computed explicitly from the addition formula for Jacobi polynomials. See [a1], Lecture 4.

For the dual problem one has

$$ P _ {n} ^ {( \alpha , \beta ) } ( x) P _ {m} ^ {( \alpha , \beta ) } ( x) = \ \sum _ {k = | n - m | } ^ { {n } + m } C ( k, m, n) P _ {k} ^ {( \alpha , \beta ) } ( x) , $$

with $ C ( k, m, n) \geq 0 $ if $ \alpha \geq \beta > - 1 $, $ \alpha + \beta \geq - 1 $. This yields a positive dual convolution structure for Jacobi series. See [a1], Lecture 5.

Jacobi polynomials admit many different group-theoretic interpretations. The three most important ones are as matrix elements of the irreducible representations of $ \mathop{\rm SU} ( 2) $( cf. [a5], Chapt. 3), as $ O ( p) \times O ( q) $- invariant spherical harmonics on the unit sphere in $ \mathbf R ^ {p + 1 } $( cf. [a2]) and as zonal spherical functions on the compact symmetric spaces of rank one (cf. [a3], Chapt. 5, §4.3).

References

[a1] R. Askey, "Orthogonal polynomials and special functions" , Reg. Conf. Ser. Appl. Math. , 21 , SIAM (1975)
[a2] B.L.J. Braaksma, B. Meulenbeld, "Jacobi polynomials as spherical harmonics" Nederl. Akad. Wetensch. Proc. Ser. A , 71 (1968) pp. 384–389
[a3] S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4
[a4] G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)
[a5] N.Ya. Vilenkin, "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian)
How to Cite This Entry:
Jacobi polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jacobi_polynomials&oldid=47459
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article