Positive quadrature formulas III: asymptotics of weights

Peherstorfer, Franz

doi:10.1090/S0025-5718-08-02119-4

Positive quadrature formulas III: asymptotics of weights
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Math. Comp. 77 (2008), 2241-2259 Request permission

Abstract:

First we discuss briefly our former characterization theorem for positive interpolation quadrature formulas (abbreviated qf), provide an equivalent characterization in terms of Jacobi matrices, and give links and applications to other qf, in particular to Gauss-Kronrod quadratures and recent rediscoveries. Then for any polynomial $t_n$ which generates a positive qf, a weight function (depending on $n$) is given with respect to which $t_n$ is orthogonal to $\mathbb {P}_{n-1}$. With the help of this result an asymptotic representation of the quadrature weights is derived. In general the asymptotic behaviour is different from that of the Gaussian weights. Only under additional conditions do the quadrature weights satisfy the so-called circle law. Corresponding results are obtained for positive qf of Radau and Lobatto type.

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