Stieltjes polynomials

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[edit]