Kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients
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 precisi...on. For this very desirable kind of extension there do not exist any results in the theory of standard quadrature formulas.
Keywords:
orthogonal polynomials / Numerical integration / Gaussian quadratures / Fourier coefficientsSource:
Mathematics of Computation, 2014, 83, 287, 1207-1231Funding / projects:
- Approximation of integral and differential operators and applications (RS-MESTD-Basic Research (BR or ON)-174015)
- Methods of Numerical and Nonlinear Analysis with Applications (RS-MESTD-Basic Research (BR or ON)-174002)
DOI: 10.1090/S0025-5718-2013-02761-5
ISSN: 0025-5718
WoS: 000337229400008
Scopus: 2-s2.0-84894728863
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Institution/Community
Mašinski fakultetTY - JOUR AU - Milovanović, Gradimir V. AU - Spalević, Miodrag PY - 2014 UR - https://machinery.mas.bg.ac.rs/handle/123456789/1951 AB - 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. T2 - Mathematics of Computation T1 - Kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients EP - 1231 IS - 287 SP - 1207 VL - 83 DO - 10.1090/S0025-5718-2013-02761-5 ER -
@article{ author = "Milovanović, Gradimir V. and Spalević, Miodrag", year = "2014", 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.", journal = "Mathematics of Computation", title = "Kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients", pages = "1231-1207", number = "287", volume = "83", doi = "10.1090/S0025-5718-2013-02761-5" }
Milovanović, G. V.,& Spalević, M.. (2014). Kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients. in Mathematics of Computation, 83(287), 1207-1231. https://doi.org/10.1090/S0025-5718-2013-02761-5
Milovanović GV, Spalević M. Kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients. in Mathematics of Computation. 2014;83(287):1207-1231. doi:10.1090/S0025-5718-2013-02761-5 .
Milovanović, Gradimir V., Spalević, Miodrag, "Kronrod extensions with multiple nodes of quadrature formulas for fourier coefficients" in Mathematics of Computation, 83, no. 287 (2014):1207-1231, https://doi.org/10.1090/S0025-5718-2013-02761-5 . .