Positive trigonometric quadrature formulas and quadrature on the unit circle

Peherstorfer, \fbox{Franz }

doi:10.1090/S0025-5718-2011-02414-2

Positive trigonometric quadrature formulas and quadrature on the unit circle
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Math. Comp. 80 (2011), 1685-1701 Request permission

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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