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Kronrod extensions with multiple nodes of quadrature formulas for Fourier coefficients

Authors: Gradimir V. Milovanović and Miodrag M. Spalević
Journal: Math. Comp. 83 (2014), 1207-1231
MSC (2010): Primary 41A55; Secondary 65D30, 65D32
Published electronically: August 28, 2013
MathSciNet review: 3167456
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Abstract: We continue with analyzing quadrature formulas of high degree of precision for computing the Fourier coefficients in expansions of functions with respect to a system of orthogonal polynomials, started recently by Bojanov and Petrova [Quadrature formulae for Fourier coefficients, J. Comput. Appl. Math. 231 (2009), 378-391] and we extend their results. Construction of new Gaussian quadrature formulas for the Fourier coefficients of a function, based on the values of the function and its derivatives, is considered. We prove the existence and uniqueness of Kronrod extensions with multiple nodes of standard Gaussian quadrature formulas with multiple nodes for several weight functions, in order to construct some new generalizations of quadrature formulas for the Fourier coefficients. For the quadrature formulas for the Fourier coefficients based on the zeros of the corresponding orthogonal polynomials we construct Kronrod extensions with multiple nodes and highest algebraic degree of precision. For this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas.

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Additional Information

Gradimir V. Milovanović
Affiliation: Mathematical Institute, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia

Miodrag M. Spalević
Affiliation: Department of Mathematics, University of Beograd, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade 35, Serbia

Keywords: Numerical integration, orthogonal polynomials, Gaussian quadratures, Fourier coefficients
Received by editor(s): February 1, 2012
Received by editor(s) in revised form: August 9, 2012, and September 4, 2012
Published electronically: August 28, 2013
Additional Notes: This work was supported in part by the Serbian Ministry of Education and Science (Research Projects: “Approximation of integral and differential operators and applications” (#174015) & “Methods of numerical and nonlinear analysis with applications” (#174002)).
Dedicated: Dedicated to the Memory of Professor Borislav Bojanov (1944–2009)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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