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A system of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202501.png" /> which satisfy the orthogonality condition
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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/s/s120/s120250/s1202502.png" /></td> </tr></table>
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where the weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202503.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202505.png" />, with finite moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202506.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202507.png" /> is the system of [[Orthogonal polynomials|orthogonal polynomials]] associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202508.png" />. The degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s1202509.png" /> is equal to its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025010.png" />. The orthogonality conditions define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025011.png" /> up to a multiplicative constant, but the conditions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025012.png" /> given above are not sufficient for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025013.png" /> to have real zeros in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025014.png" />. However, several special cases and classes of weight functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025015.png" /> are known for which the zeros of the corresponding Stieltjes polynomials do not only have this property, but also interlace with the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025016.png" />. A simple example is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025019.png" />, the Chebyshev polynomial of the second kind (cf. [[Chebyshev polynomials|Chebyshev polynomials]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025020.png" />, the Chebyshev polynomial of the first kind. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025023.png" />. A generalization are the Bernstein–Szegö weight functions
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A system of polynomials $\{ E _ { n  + 1} \}$ which satisfy the orthogonality condition
  
<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/s/s120/s120250/s12025024.png" /></td> </tr></table>
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\begin{equation*} \int _ { a } ^ { b } P _ { n } ( x ) E _ { n + 1 } ( x ) x ^ { k } h ( x ) d x = 0 , \quad k = 1 , \dots , n, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025025.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025026.png" /> that is positive in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025027.png" /> [[#References|[a6]]], [[#References|[a7]]]. Weight functions for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025031.png" /> are another class for which the above properties are known to hold asymptotically under certain additional conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025032.png" /> [[#References|[a8]]].
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where the weight function $h$ satisfies $h \geq 0$, $0 &lt; \int _ { a } ^ { b } h ( x ) d x &lt; \infty$, with finite moments $h _ { n } = \int _ { a } ^ { b } x ^ { n } h ( x ) d x$. $\{ P _ { n } \}$ is the system of [[Orthogonal polynomials|orthogonal polynomials]] associated with $h$. The degree of $E _ { n  + 1}$ is equal to its index $n + 1$. The orthogonality conditions define $E _ { n  + 1}$ up to a multiplicative constant, but the conditions for $h$ given above are not sufficient for $E _ { n  + 1}$ to have real zeros in $[ a , b ]$. However, several special cases and classes of weight functions $h$ are known for which the zeros of the corresponding Stieltjes polynomials do not only have this property, but also interlace with the zeros of $P_n$. A simple example is $[ a , b ] = [ - 1,1 ]$, $h ( x ) = \sqrt { 1 - x ^ { 2 } }$, $P _ { n } ( x ) = U _ { n } ( x )$, the Chebyshev polynomial of the second kind (cf. [[Chebyshev polynomials|Chebyshev polynomials]]), where $E _ { n + 1 } ( x ) = T _ { n + 1 } ( x )$, the Chebyshev polynomial of the first kind. For $h ( x ) = ( 1 - x ^ { 2 } ) ^ { - 1 / 2 }$, $P _ { n } ( x ) = T _ { n } ( x )$ and $E _ { n + 1 } ( x ) = ( 1 - x ^ { 2 } ) U _ { n - 1 } ( x )$. A generalization are the Bernstein–Szegö weight functions
  
The classical case originally considered by Th.J. Stieltjes is the Legendre weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025034.png" />. For this case G. Szegö [[#References|[a9]]] proved that all zeros are real, belong to the open interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025035.png" /> and interlace with the zeros of the [[Legendre polynomials|Legendre polynomials]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025036.png" />. Szegö extended his proof to the ultraspherical, or Gegenbauer, weight function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025039.png" />, cf. also [[Gegenbauer polynomials|Gegenbauer polynomials]]; [[Ultraspherical polynomials|Ultraspherical polynomials]]. For the more general Jacobi weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025040.png" />, results of existence and non-existence can be found in [[#References|[a4]]]. Comparatively little is known for unbounded intervals. Numerical results reported in [[#References|[a5]]] show that complex zeros arise for the Laguerre weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025043.png" /> and the Hermite weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025045.png" />.
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\begin{equation*} h ( x ) = \frac { ( 1 - x ^ { 2 } ) ^ { \pm 1 / 2 } } { \rho _ { m } ( x ) }, \end{equation*}
  
An important fact for the analysis of the Stieltjes polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025046.png" /> is their close connection with the functions of the second kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025047.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025049.png" />. Stieltjes [[#References|[a1]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025050.png" /> is precisely the polynomial part of the Laurent expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025051.png" />. Szegö's work in [[#References|[a9]]] and subsequent investigations are based on this connection.
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where $\rho _ { m }$ is a polynomial of degree $m$ that is positive in $[ - 1,1 ]$ [[#References|[a6]]], [[#References|[a7]]]. Weight functions for which $\operatorname { log } h / \sqrt { 1 - x ^ { 2 } } \in L _ { 1 } [ - 1,1 ]$, $\sqrt { 1 - x ^ { 2 } } h \in C [ - 1,1 ]$ and $\sqrt { 1 - x ^ { 2 } } w ( x ) &gt; 0$ for $x \in [ - 1,1 ]$ are another class for which the above properties are known to hold asymptotically under certain additional conditions on $h$ [[#References|[a8]]].
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The classical case originally considered by Th.J. Stieltjes is the Legendre weight function $ { h } \equiv 1$, $[ a , b ] = [ - 1,1 ]$. For this case G. Szegö [[#References|[a9]]] proved that all zeros are real, belong to the open interval $( - 1,1 )$ and interlace with the zeros of the [[Legendre polynomials|Legendre polynomials]] $P_n$. Szegö extended his proof to the ultraspherical, or Gegenbauer, weight function, $h ( x ) = ( 1 - x ^ { 2 } ) ^ { \lambda - 1 / 2 }$, $\lambda \in [ 0,2 ]$, $[ a , b ] = [ - 1,1 ]$, cf. also [[Gegenbauer polynomials|Gegenbauer polynomials]]; [[Ultraspherical polynomials|Ultraspherical polynomials]]. For the more general Jacobi weight $h ( x ) = ( 1 - x ) ^ { \alpha } ( 1 + x ) ^ { \beta }$, results of existence and non-existence can be found in [[#References|[a4]]]. Comparatively little is known for unbounded intervals. Numerical results reported in [[#References|[a5]]] show that complex zeros arise for the Laguerre weight $h ( x ) = x ^ { \alpha } \operatorname { exp } ( - x )$, $\alpha &gt; - 1$, $( a , b ) = ( 0 , \infty )$ and the Hermite weight $h ( x ) = \operatorname { exp } ( - x ^ { 2 } )$, $( a , b ) = ( - \infty , \infty )$.
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An important fact for the analysis of the Stieltjes polynomials $E _ { n  + 1}$ is their close connection with the functions of the second kind $Q _ { n }$ associated with $P_n$ and $h$. Stieltjes [[#References|[a1]]] proved that $E _ { n  + 1}$ is precisely the polynomial part of the Laurent expansion of $[ Q _ { n } ] ^ { - 1 }$. Szegö's work in [[#References|[a9]]] and subsequent investigations are based on this connection.
  
 
Several asymptotic representations are available. A simple formula for the Legendre weight function is
 
Several asymptotic representations are available. A simple formula for the Legendre weight function is
  
<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/s/s120/s120250/s12025052.png" /></td> </tr></table>
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\begin{equation*} E _ { n  + 1} ( \operatorname { cos } \theta ) = \end{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/s/s120/s120250/s12025053.png" /></td> </tr></table>
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\begin{equation*} = 2 \left( \frac { 2 n \operatorname { sin } \theta } { \pi } \right) ^ { 1 / 2 } \operatorname { cos } \left\{ \left( n + \frac { 1 } { 2 } \right) \theta + \frac { \pi } { 4 } \right\} + \mathcal{O} ( 1 ), \end{equation*}
  
uniformly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025054.png" />, see [[#References|[a2]]]. Inequalities for Stieltjes polynomials in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025055.png" /> can be found in [[#References|[a3]]].
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uniformly for $\epsilon \leq \theta \leq \pi - \epsilon$, see [[#References|[a2]]]. Inequalities for Stieltjes polynomials in the case $ { h } \equiv 1$ can be found in [[#References|[a3]]].
  
The zeros of Stieltjes polynomials are used for quadrature and for interpolation. In particular, the often-used Gauss–Kronrod quadrature formulas (cf. [[Gauss–Kronrod quadrature formula|Gauss–Kronrod quadrature formula]]) are based on the union of the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025057.png" /> and enable an efficient estimation for the [[Gauss quadrature formula|Gauss quadrature formula]] based on the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025058.png" />. This idea has been carried over to extended interpolation processes (cf. [[Extended interpolation process|Extended interpolation process]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025059.png" />, adding the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025060.png" /> improves the interpolation process based on the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120250/s12025061.png" /> to an optimal-order interpolation process [[#References|[a3]]] (see also [[Extended interpolation process|Extended interpolation process]]).
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The zeros of Stieltjes polynomials are used for quadrature and for interpolation. In particular, the often-used Gauss–Kronrod quadrature formulas (cf. [[Gauss–Kronrod quadrature formula|Gauss–Kronrod quadrature formula]]) are based on the union of the zeros of $P_n$ and $E _ { n  + 1}$ and enable an efficient estimation for the [[Gauss quadrature formula|Gauss quadrature formula]] based on the zeros of $P_n$. This idea has been carried over to extended interpolation processes (cf. [[Extended interpolation process|Extended interpolation process]]). For $ { h } \equiv 1$, adding the zeros of $E _ { n  + 1}$ improves the interpolation process based on the zeros of $P_n$ to an optimal-order interpolation process [[#References|[a3]]] (see also [[Extended interpolation process|Extended interpolation process]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Baillaud,  H. Bourget,  "Correspondance d'Hermite et de Stieltjes" , '''I,II''' , Gauthier-Villars  (1905)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Ehrich,  "Asymptotic properties of Stieltjes polynomials and Gauss–Kronrod quadrature formulae"  ''J. Approx. Th.'' , '''82'''  (1995)  pp. 287–303</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Ehrich,  G. Mastroianni,  "Stieltjes polynomials and Lagrange interpolation"  ''Math. Comput.'' , '''66'''  (1997)  pp. 311–331</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Gautschi,  S.E. Notaris,  "An algebraic study of Gauss–Kronrod quadrature formulae for Jacobi weight functions"  ''Math. Comput.'' , '''51'''  (1988)  pp. 231–248</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G. Monegato,  "Stieltjes polynomials and related quadrature rules"  ''SIAM Review'' , '''24'''  (1982)  pp. 137–158</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S.E. Notaris,  "Gauss–Kronrod quadrature for weight functions of Bernstein–Szegö type"  ''J. Comput. Appl. Math.'' , '''29'''  (1990)  pp. 161–169</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F. Peherstorfer,  "Weight functions admitting repeated positive Kronrod quadrature"  ''BIT'' , '''30'''  (1990)  pp. 241–251</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  F. Peherstorfer,  "Stieltjes polynomials and functions of the second kind"  ''J. Comput. Appl. Math.'' , '''65'''  (1995)  pp. 319–338</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Szegö,  "Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören"  ''Math. Ann.'' , '''110'''  (1934)  pp. 501–513  (Collected papers, Vol.2, R. Askey (Ed.), Birkhäuser, 1982, 545-557)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  B. Baillaud,  H. Bourget,  "Correspondance d'Hermite et de Stieltjes" , '''I,II''' , Gauthier-Villars  (1905)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Ehrich,  "Asymptotic properties of Stieltjes polynomials and Gauss–Kronrod quadrature formulae"  ''J. Approx. Th.'' , '''82'''  (1995)  pp. 287–303</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  S. Ehrich,  G. Mastroianni,  "Stieltjes polynomials and Lagrange interpolation"  ''Math. Comput.'' , '''66'''  (1997)  pp. 311–331</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  W. Gautschi,  S.E. Notaris,  "An algebraic study of Gauss–Kronrod quadrature formulae for Jacobi weight functions"  ''Math. Comput.'' , '''51'''  (1988)  pp. 231–248</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  G. Monegato,  "Stieltjes polynomials and related quadrature rules"  ''SIAM Review'' , '''24'''  (1982)  pp. 137–158</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  S.E. Notaris,  "Gauss–Kronrod quadrature for weight functions of Bernstein–Szegö type"  ''J. Comput. Appl. Math.'' , '''29'''  (1990)  pp. 161–169</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  F. Peherstorfer,  "Weight functions admitting repeated positive Kronrod quadrature"  ''BIT'' , '''30'''  (1990)  pp. 241–251</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  F. Peherstorfer,  "Stieltjes polynomials and functions of the second kind"  ''J. Comput. Appl. Math.'' , '''65'''  (1995)  pp. 319–338</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  G. Szegö,  "Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören"  ''Math. Ann.'' , '''110'''  (1934)  pp. 501–513  (Collected papers, Vol.2, R. Askey (Ed.), Birkhäuser, 1982, 545-557)</td></tr></table>

Revision as of 17:00, 1 July 2020

A system of polynomials $\{ E _ { n + 1} \}$ which satisfy the orthogonality condition

\begin{equation*} \int _ { a } ^ { b } P _ { n } ( x ) E _ { n + 1 } ( x ) x ^ { k } h ( x ) d x = 0 , \quad k = 1 , \dots , n, \end{equation*}

where the weight function $h$ satisfies $h \geq 0$, $0 < \int _ { a } ^ { b } h ( x ) d x < \infty$, with finite moments $h _ { n } = \int _ { a } ^ { b } x ^ { n } h ( x ) d x$. $\{ P _ { n } \}$ is the system of orthogonal polynomials associated with $h$. The degree of $E _ { n + 1}$ is equal to its index $n + 1$. The orthogonality conditions define $E _ { n + 1}$ up to a multiplicative constant, but the conditions for $h$ given above are not sufficient for $E _ { n + 1}$ to have real zeros in $[ a , b ]$. However, several special cases and classes of weight functions $h$ are known for which the zeros of the corresponding Stieltjes polynomials do not only have this property, but also interlace with the zeros of $P_n$. A simple example is $[ a , b ] = [ - 1,1 ]$, $h ( x ) = \sqrt { 1 - x ^ { 2 } }$, $P _ { n } ( x ) = U _ { n } ( x )$, the Chebyshev polynomial of the second kind (cf. Chebyshev polynomials), where $E _ { n + 1 } ( x ) = T _ { n + 1 } ( x )$, the Chebyshev polynomial of the first kind. For $h ( x ) = ( 1 - x ^ { 2 } ) ^ { - 1 / 2 }$, $P _ { n } ( x ) = T _ { n } ( x )$ and $E _ { n + 1 } ( x ) = ( 1 - x ^ { 2 } ) U _ { n - 1 } ( x )$. A generalization are the Bernstein–Szegö weight functions

\begin{equation*} h ( x ) = \frac { ( 1 - x ^ { 2 } ) ^ { \pm 1 / 2 } } { \rho _ { m } ( x ) }, \end{equation*}

where $\rho _ { m }$ is a polynomial of degree $m$ that is positive in $[ - 1,1 ]$ [a6], [a7]. Weight functions for which $\operatorname { log } h / \sqrt { 1 - x ^ { 2 } } \in L _ { 1 } [ - 1,1 ]$, $\sqrt { 1 - x ^ { 2 } } h \in C [ - 1,1 ]$ and $\sqrt { 1 - x ^ { 2 } } w ( x ) > 0$ for $x \in [ - 1,1 ]$ are another class for which the above properties are known to hold asymptotically under certain additional conditions on $h$ [a8].

The classical case originally considered by Th.J. Stieltjes is the Legendre weight function $ { h } \equiv 1$, $[ a , b ] = [ - 1,1 ]$. For this case G. Szegö [a9] proved that all zeros are real, belong to the open interval $( - 1,1 )$ and interlace with the zeros of the Legendre polynomials $P_n$. Szegö extended his proof to the ultraspherical, or Gegenbauer, weight function, $h ( x ) = ( 1 - x ^ { 2 } ) ^ { \lambda - 1 / 2 }$, $\lambda \in [ 0,2 ]$, $[ a , b ] = [ - 1,1 ]$, cf. also Gegenbauer polynomials; Ultraspherical polynomials. For the more general Jacobi weight $h ( x ) = ( 1 - x ) ^ { \alpha } ( 1 + x ) ^ { \beta }$, results of existence and non-existence can be found in [a4]. Comparatively little is known for unbounded intervals. Numerical results reported in [a5] show that complex zeros arise for the Laguerre weight $h ( x ) = x ^ { \alpha } \operatorname { exp } ( - x )$, $\alpha > - 1$, $( a , b ) = ( 0 , \infty )$ and the Hermite weight $h ( x ) = \operatorname { exp } ( - x ^ { 2 } )$, $( a , b ) = ( - \infty , \infty )$.

An important fact for the analysis of the Stieltjes polynomials $E _ { n + 1}$ is their close connection with the functions of the second kind $Q _ { n }$ associated with $P_n$ and $h$. Stieltjes [a1] proved that $E _ { n + 1}$ is precisely the polynomial part of the Laurent expansion of $[ Q _ { n } ] ^ { - 1 }$. Szegö's work in [a9] and subsequent investigations are based on this connection.

Several asymptotic representations are available. A simple formula for the Legendre weight function is

\begin{equation*} E _ { n + 1} ( \operatorname { cos } \theta ) = \end{equation*}

\begin{equation*} = 2 \left( \frac { 2 n \operatorname { sin } \theta } { \pi } \right) ^ { 1 / 2 } \operatorname { cos } \left\{ \left( n + \frac { 1 } { 2 } \right) \theta + \frac { \pi } { 4 } \right\} + \mathcal{O} ( 1 ), \end{equation*}

uniformly for $\epsilon \leq \theta \leq \pi - \epsilon$, see [a2]. Inequalities for Stieltjes polynomials in the case $ { h } \equiv 1$ can be found in [a3].

The zeros of Stieltjes polynomials are used for quadrature and for interpolation. In particular, the often-used Gauss–Kronrod quadrature formulas (cf. Gauss–Kronrod quadrature formula) are based on the union of the zeros of $P_n$ and $E _ { n + 1}$ and enable an efficient estimation for the Gauss quadrature formula based on the zeros of $P_n$. This idea has been carried over to extended interpolation processes (cf. Extended interpolation process). For $ { h } \equiv 1$, adding the zeros of $E _ { n + 1}$ improves the interpolation process based on the zeros of $P_n$ to an optimal-order interpolation process [a3] (see also Extended interpolation process).

References

[a1] B. Baillaud, H. Bourget, "Correspondance d'Hermite et de Stieltjes" , I,II , Gauthier-Villars (1905)
[a2] S. Ehrich, "Asymptotic properties of Stieltjes polynomials and Gauss–Kronrod quadrature formulae" J. Approx. Th. , 82 (1995) pp. 287–303
[a3] S. Ehrich, G. Mastroianni, "Stieltjes polynomials and Lagrange interpolation" Math. Comput. , 66 (1997) pp. 311–331
[a4] W. Gautschi, S.E. Notaris, "An algebraic study of Gauss–Kronrod quadrature formulae for Jacobi weight functions" Math. Comput. , 51 (1988) pp. 231–248
[a5] G. Monegato, "Stieltjes polynomials and related quadrature rules" SIAM Review , 24 (1982) pp. 137–158
[a6] S.E. Notaris, "Gauss–Kronrod quadrature for weight functions of Bernstein–Szegö type" J. Comput. Appl. Math. , 29 (1990) pp. 161–169
[a7] F. Peherstorfer, "Weight functions admitting repeated positive Kronrod quadrature" BIT , 30 (1990) pp. 241–251
[a8] F. Peherstorfer, "Stieltjes polynomials and functions of the second kind" J. Comput. Appl. Math. , 65 (1995) pp. 319–338
[a9] G. Szegö, "Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören" Math. Ann. , 110 (1934) pp. 501–513 (Collected papers, Vol.2, R. Askey (Ed.), Birkhäuser, 1982, 545-557)
How to Cite This Entry:
Stieltjes polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stieltjes_polynomials&oldid=14373
This article was adapted from an original article by Sven Ehrich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article