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''spherical polynomials''
 
''spherical polynomials''
  
Polynomials orthogonal on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580501.png" /> with unit weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l0580502.png" />. The standardized Legendre polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]
+
Polynomials orthogonal on the interval $  [ - 1 , 1 ] $
 +
with unit weight $  \phi ( x) = 1 $.  
 +
The standardized Legendre polynomials are defined by the [[Rodrigues formula|Rodrigues formula]]
 +
 
 +
$$
 +
P _ {n} ( 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/l/l058/l058050/l0580503.png" /></td> </tr></table>
+
\frac{1}{n ! 2  ^ {n} }
 +
 
 +
\frac{d  ^ {n} }{d x  ^ {n} }
 +
 
 +
( x  ^ {2} - 1 )  ^ {n} ,\ \
 +
n = 0 , 1 \dots
 +
$$
  
 
and have the representation
 
and have the representation
  
<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/l/l058/l058050/l0580504.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( x)  = \
 +
 
 +
\frac{1}{2  ^ {n} }
 +
 
 +
\sum _ { k= } 0 ^ { [ }  n/2]
 +
 
 +
\frac{( - 1 )  ^ {k} ( 2 n - 2 k ) ! }{k ! ( n - k ) ! ( n - 2 k ) ! }
 +
 
 +
x  ^ {n-} 2k .
 +
$$
  
 
The formulas most commonly used are:
 
The formulas most commonly used 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/l/l058/l058050/l0580505.png" /></td> </tr></table>
+
$$
 +
( n + 1 ) P _ {n+} 1 ( x)  = \
 +
( 2 n + 1 ) x P _ {n} ( x) - n P _ {n-} 1 ( x) ,
 +
$$
  
<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/l/l058/l058050/l0580506.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( - x )  = ( - 1 )  ^ {n} P _ {n} ( x) ; \  P _ {n} ( 1)  = 1 ,\  P _ {n} ( - 1 )  = ( - 1 )  ^ {n} ,
 +
$$
  
<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/l/l058/l058050/l0580507.png" /></td> </tr></table>
+
$$
 +
( 1 - x  ^ {2} ) P _ {n} ^ { \prime } ( x)  = n P _ {n-} 1 ( x) - x n P _ {n} ( x) ,
 +
$$
  
<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/l/l058/l058050/l0580508.png" /></td> </tr></table>
+
$$
 +
P _ {n+} 1 ^ { \prime } ( x) - P _ {n-} 1 ^ { \prime } ( x)  = ( 2 n + 1 ) P _ {n} ( x) .
 +
$$
  
 
The Legendre polynomials can be defined as the coefficients in the expansion of the generating function
 
The Legendre polynomials can be defined as the coefficients in the expansion of the generating function
  
<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/l/l058/l058050/l0580509.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{1}{\sqrt {1 - 2 x t + t  ^ {2} }}
 +
  = \
 +
\sum _ { n= } 0 ^  \infty 
 +
P _ {n} ( x) t  ^ {n} ,
 +
$$
  
where the series on the right-hand side converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805010.png" />.
+
where the series on the right-hand side converges for $  x \in [ - 1 , 1 ] $.
  
 
The first few standardized Legendre polynomials have the form
 
The first few standardized Legendre 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/l/l058/l058050/l05805011.png" /></td> </tr></table>
+
$$
 +
P _ {0} ( x)  = 1 ,\  P _ {1} ( x)  = x ,\ \
 +
P _ {2} ( x)  =
 +
\frac{3 x  ^ {2} - 1 }{2}
 +
,
 +
$$
 +
 
 +
$$
 +
P _ {3} ( x)  =
 +
\frac{5 x  ^ {3} - 3 x }{2}
 +
,\  P _ {4} ( x)  =
 +
\frac{35 x  ^ {4} - 30 x  ^ {2} + 3 }{8}
 +
,
 +
$$
  
<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/l/l058/l058050/l05805012.png" /></td> </tr></table>
+
$$
 +
P _ {5} ( x)  =
 +
\frac{63 x  ^ {5} - 70 x  ^ {3} + 15 x }{8}
 +
,
 +
$$
  
<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/l/l058/l058050/l05805013.png" /></td> </tr></table>
+
$$
 +
P _ {6} ( x)  =
 +
\frac{231 x  ^ {6} - 315 x  ^ {4} + 105 x  ^ {2} - 5 }{16}
 +
.
 +
$$
  
<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/l/l058/l058050/l05805014.png" /></td> </tr></table>
+
The Legendre polynomial of order  $  n $
 +
satisfies the differential equation (Legendre equation)
  
The Legendre polynomial of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805015.png" /> satisfies the differential equation (Legendre equation)
+
$$
 +
( 1 - x  ^ {2} )
  
<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/l/l058/l058050/l05805016.png" /></td> </tr></table>
+
\frac{d  ^ {2} y }{d x  ^ {2} }
 +
-
 +
2 x
 +
\frac{dy}{dx}
 +
+ n
 +
( n + 1 ) y  = 0 ,
 +
$$
  
 
which occurs in the solution of the [[Laplace equation|Laplace equation]] in spherical coordinates by the method of separation of variables. The orthogonal Legendre polynomials have the form
 
which occurs in the solution of the [[Laplace equation|Laplace equation]] in spherical coordinates by the method of separation of variables. The orthogonal Legendre 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/l/l058/l058050/l05805017.png" /></td> </tr></table>
+
$$
 +
\widehat{P}  _ {n} ( x)  = \
 +
\sqrt {
 +
\frac{2 n + 1 }{2}
 +
}
 +
P _ {n} ( x) ,\  n = 0 , 1 \dots
 +
$$
  
 
and satisfy the uniform and weighted estimates
 
and satisfy the uniform and weighted estimates
  
<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/l/l058/l058050/l05805018.png" /></td> </tr></table>
+
$$
 +
| \widehat{P}  _ {n} ( x) |  \leq  \
 +
\sqrt {
 +
\frac{2 n + 1 }{2}
 +
} ,\ \
 +
x \in [ - 1 , 1 ] ,
 +
$$
  
<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/l/l058/l058050/l05805019.png" /></td> </tr></table>
+
$$
 +
( 1 - x  ^ {2} )  ^ {1/4} | \widehat{P}  _ {n} ( x) |  \leq  \sqrt {
 +
\frac{2
 +
n + 1 }{\pi n }
 +
} ,\  x \in [ - 1 , 1 ] .
 +
$$
  
Fourier series in the Legendre polynomials inside the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805020.png" /> are analogous to trigonometric [[Fourier series|Fourier series]] (cf. also [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]); there is a theorem about the equiconvergence of these two series, which implies that the Fourier–Legendre series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805021.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805022.png" /> converges if and only if the trigonometric Fourier series of the function
+
Fourier series in the Legendre polynomials inside the interval $  ( - 1 , 1 ) $
 +
are analogous to trigonometric [[Fourier series|Fourier series]] (cf. also [[Fourier series in orthogonal polynomials|Fourier series in orthogonal polynomials]]); there is a theorem about the equiconvergence of these two series, which implies that the Fourier–Legendre series of a function $  f $
 +
at a point $  x \in ( - 1 , 1 ) $
 +
converges if and only if the trigonometric Fourier series of the function
  
<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/l/l058/l058050/l05805023.png" /></td> </tr></table>
+
$$
 +
F ( \theta )  = \
 +
( \sin  \theta )  ^ {1/2}
 +
f ( \cos  \theta )
 +
$$
  
converges at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805024.png" />. In a neighbourhood of the end points the situation is different, since the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805025.png" /> increases with speed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805027.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805028.png" /> and satisfies a [[Lipschitz condition|Lipschitz condition]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805029.png" />, then the Fourier–Legendre series converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805030.png" /> uniformly on the whole interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805032.png" />, then this series generally diverges at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805033.png" />.
+
converges at the point $  \theta = { \mathop{\rm arc}  \cos }  x $.  
 +
In a neighbourhood of the end points the situation is different, since the sequence $  \{ \widehat{P}  _ {n} ( \pm  1 ) \} $
 +
increases with speed $  \sqrt n $.  
 +
If $  f $
 +
is continuous on $  [ - 1 , 1 ] $
 +
and satisfies a [[Lipschitz condition|Lipschitz condition]] of order $  \alpha > 1 / 2 $,  
 +
then the Fourier–Legendre series converges to $  f $
 +
uniformly on the whole interval $  [ - 1 , 1 ] $.  
 +
If $  \alpha = 1 / 2 $,  
 +
then this series generally diverges at the points $  x = \pm  1 $.
  
 
These polynomials were introduced by A.M. Legendre [[#References|[1]]].
 
These polynomials were introduced by A.M. Legendre [[#References|[1]]].
Line 61: Line 171:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Legendre,  ''Mém. Math. Phys. présentés à l'Acad. Sci. par divers savants'' , '''10'''  (1785)  pp. 411–434</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.W. Hobson,  "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press  (1931)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Legendre,  ''Mém. Math. Phys. présentés à l'Acad. Sci. par divers savants'' , '''10'''  (1785)  pp. 411–434</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.W. Hobson,  "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press  (1931)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Legendre polynomials belong to the families of [[Gegenbauer polynomials|Gegenbauer polynomials]]; [[Jacobi polynomials|Jacobi polynomials]] and [[Classical orthogonal polynomials|classical orthogonal polynomials]]. They can be written as hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]). Their group-theoretic interpretation as [[Zonal spherical functions|zonal spherical functions]] on the two-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058050/l05805034.png" /> serves as a prototype, both from the historical and the didactical point of view. A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.
+
Legendre polynomials belong to the families of [[Gegenbauer polynomials|Gegenbauer polynomials]]; [[Jacobi polynomials|Jacobi polynomials]] and [[Classical orthogonal polynomials|classical orthogonal polynomials]]. They can be written as hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]). Their group-theoretic interpretation as [[Zonal spherical functions|zonal spherical functions]] on the two-dimensional sphere $  S  ^ {2} = \mathop{\rm SO} ( 3) / \mathop{\rm SO} ( 2) $
 +
serves as a prototype, both from the historical and the didactical point of view. A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.

Latest revision as of 22:16, 5 June 2020


spherical polynomials

Polynomials orthogonal on the interval $ [ - 1 , 1 ] $ with unit weight $ \phi ( x) = 1 $. The standardized Legendre polynomials are defined by the Rodrigues formula

$$ P _ {n} ( x) = \ \frac{1}{n ! 2 ^ {n} } \frac{d ^ {n} }{d x ^ {n} } ( x ^ {2} - 1 ) ^ {n} ,\ \ n = 0 , 1 \dots $$

and have the representation

$$ P _ {n} ( x) = \ \frac{1}{2 ^ {n} } \sum _ { k= } 0 ^ { [ } n/2] \frac{( - 1 ) ^ {k} ( 2 n - 2 k ) ! }{k ! ( n - k ) ! ( n - 2 k ) ! } x ^ {n-} 2k . $$

The formulas most commonly used are:

$$ ( n + 1 ) P _ {n+} 1 ( x) = \ ( 2 n + 1 ) x P _ {n} ( x) - n P _ {n-} 1 ( x) , $$

$$ P _ {n} ( - x ) = ( - 1 ) ^ {n} P _ {n} ( x) ; \ P _ {n} ( 1) = 1 ,\ P _ {n} ( - 1 ) = ( - 1 ) ^ {n} , $$

$$ ( 1 - x ^ {2} ) P _ {n} ^ { \prime } ( x) = n P _ {n-} 1 ( x) - x n P _ {n} ( x) , $$

$$ P _ {n+} 1 ^ { \prime } ( x) - P _ {n-} 1 ^ { \prime } ( x) = ( 2 n + 1 ) P _ {n} ( x) . $$

The Legendre polynomials can be defined as the coefficients in the expansion of the generating function

$$ \frac{1}{\sqrt {1 - 2 x t + t ^ {2} }} = \ \sum _ { n= } 0 ^ \infty P _ {n} ( x) t ^ {n} , $$

where the series on the right-hand side converges for $ x \in [ - 1 , 1 ] $.

The first few standardized Legendre polynomials have the form

$$ P _ {0} ( x) = 1 ,\ P _ {1} ( x) = x ,\ \ P _ {2} ( x) = \frac{3 x ^ {2} - 1 }{2} , $$

$$ P _ {3} ( x) = \frac{5 x ^ {3} - 3 x }{2} ,\ P _ {4} ( x) = \frac{35 x ^ {4} - 30 x ^ {2} + 3 }{8} , $$

$$ P _ {5} ( x) = \frac{63 x ^ {5} - 70 x ^ {3} + 15 x }{8} , $$

$$ P _ {6} ( x) = \frac{231 x ^ {6} - 315 x ^ {4} + 105 x ^ {2} - 5 }{16} . $$

The Legendre polynomial of order $ n $ satisfies the differential equation (Legendre equation)

$$ ( 1 - x ^ {2} ) \frac{d ^ {2} y }{d x ^ {2} } - 2 x \frac{dy}{dx} + n ( n + 1 ) y = 0 , $$

which occurs in the solution of the Laplace equation in spherical coordinates by the method of separation of variables. The orthogonal Legendre polynomials have the form

$$ \widehat{P} _ {n} ( x) = \ \sqrt { \frac{2 n + 1 }{2} } P _ {n} ( x) ,\ n = 0 , 1 \dots $$

and satisfy the uniform and weighted estimates

$$ | \widehat{P} _ {n} ( x) | \leq \ \sqrt { \frac{2 n + 1 }{2} } ,\ \ x \in [ - 1 , 1 ] , $$

$$ ( 1 - x ^ {2} ) ^ {1/4} | \widehat{P} _ {n} ( x) | \leq \sqrt { \frac{2 n + 1 }{\pi n } } ,\ x \in [ - 1 , 1 ] . $$

Fourier series in the Legendre polynomials inside the interval $ ( - 1 , 1 ) $ are analogous to trigonometric Fourier series (cf. also Fourier series in orthogonal polynomials); there is a theorem about the equiconvergence of these two series, which implies that the Fourier–Legendre series of a function $ f $ at a point $ x \in ( - 1 , 1 ) $ converges if and only if the trigonometric Fourier series of the function

$$ F ( \theta ) = \ ( \sin \theta ) ^ {1/2} f ( \cos \theta ) $$

converges at the point $ \theta = { \mathop{\rm arc} \cos } x $. In a neighbourhood of the end points the situation is different, since the sequence $ \{ \widehat{P} _ {n} ( \pm 1 ) \} $ increases with speed $ \sqrt n $. If $ f $ is continuous on $ [ - 1 , 1 ] $ and satisfies a Lipschitz condition of order $ \alpha > 1 / 2 $, then the Fourier–Legendre series converges to $ f $ uniformly on the whole interval $ [ - 1 , 1 ] $. If $ \alpha = 1 / 2 $, then this series generally diverges at the points $ x = \pm 1 $.

These polynomials were introduced by A.M. Legendre [1].

See also the references to Orthogonal polynomials.

References

[1] A.M. Legendre, Mém. Math. Phys. présentés à l'Acad. Sci. par divers savants , 10 (1785) pp. 411–434
[2] E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press (1931)

Comments

Legendre polynomials belong to the families of Gegenbauer polynomials; Jacobi polynomials and classical orthogonal polynomials. They can be written as hypergeometric functions (cf. Hypergeometric function). Their group-theoretic interpretation as zonal spherical functions on the two-dimensional sphere $ S ^ {2} = \mathop{\rm SO} ( 3) / \mathop{\rm SO} ( 2) $ serves as a prototype, both from the historical and the didactical point of view. A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.

How to Cite This Entry:
Legendre polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Legendre_polynomials&oldid=47609
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article