Positive rational interpolatory quadrature formulas on the unit circle and the interval

Authors:
Karl Deckers;Adhemar Bultheel;Ruymán Cruz-Barroso;Francisco Perdomo-Pío
Affiliations:
Department of Computer Science, K.U.Leuven, Celestijnenlaan 200 A, B-3001 Heverlee, Belgium;Department of Computer Science, K.U.Leuven, Celestijnenlaan 200 A, B-3001 Heverlee, Belgium;Department of Mathematical Analysis, La Laguna University, 38271 La Laguna, Tenerife, Canary Islands, Spain;Department of Mathematical Analysis, La Laguna University, 38271 La Laguna, Tenerife, Canary Islands, Spain
Venue:
Applied Numerical Mathematics
Year:
2010

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Abstract

We present a relation between rational Gauss-type quadrature formulas that approximate integrals of the form J"@m(F)=@!"-"1^1F(x)d@m(x), and rational Szego quadrature formulas that approximate integrals of the form I"@m"@?(F)=@!"-"@p^@pF(e^i^@q)d@m@?(@q). The measures @m and @m@? are assumed to be positive bounded Borel measures on the interval [-1,1] and the complex unit circle respectively, and are related by @m@?^'(@q)=@m^'(cos@q)|sin@q|. Next, making use of the so-called para-orthogonal rational functions, we obtain a one-parameter family of rational interpolatory quadrature formulas with positive weights for J"@m(F). Finally, we include some illustrative numerical examples.