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Positive trigonometric quadrature formulas and quadrature on the unit circle


Author: Franz Peherstorfer
Journal: Math. Comp. 80 (2011), 1685-1701
MSC (2010): Primary 65D30, 41A55, 42C05
Published electronically: March 3, 2011
MathSciNet review: 2785474
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Abstract: We give several descriptions of positive quadrature formulas which are exact for trigonometric-, respectively, Laurent polynomials of degree less or equal to $ 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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Additional Information

Franz Peherstorfer
Affiliation: Group for Dynamical Systems and Approximation Theory, Institute for Analysis, Johannes Kepler University, A-4040 Linz, Austria

DOI: https://doi.org/10.1090/S0025-5718-2011-02414-2
Keywords: Trigonometric quadrature formulas, orthogonal polynomials, positive quadrature weights, recurrence relation, para-orthogonal polynomials, zeros, asymptotics
Received by editor(s): November 6, 2008
Received by editor(s) in revised form: January 29, 2010
Published electronically: March 3, 2011
Additional Notes: The author was supported by the Austrian Science Fund FWF, project no. P20413-N18
Sadly, on November 27, 2009 the author, Franz Peherstorfer, passed away. Since then, Ionela Moale, his last PhD student, did the main work in preparing the paper for publication, and Franz’s colleague, Professor Peter Yuditskii, has kindly stepped in to proofread the galley proof of the work.
Article copyright: © Copyright 2011 American Mathematical Society