Positive trigonometric Quadrature Formulas and quadrature on the unit circle

Details Positive trigonometric Quadrature Formulas and quadrature on the unit circle Positive trigonometric Quadrature Formulas and quadrature on the unit circle

2010-01-14

arxiv.org/abs/1001.2451v1

Mathematics

Classical Analysis and ODEs

Submitted by F. Peherstorfer to Math. Comp

Scientific paper

We give several descriptions of positive quadrature formulas which are exact for trigonometric -, respectively, Laurent polynomials of degree less or equal $n-1-m$, $0\leq m\leq n-1$. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a byproduct interlacing properties of zeros of para-orthogonal polynomials are obtained. Finally, asymptotics for the quadrature weights are presented.

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