Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal B-spline

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B -splines φ m as the weight functions on [0, m ] ( m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussi...

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Veröffentlicht in:	Numerical algorithms Jg. 76; H. 2; S. 333 - 347
1. Verfasser:	Milovanović, Gradimir V.
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.10.2017 Springer Nature B.V
Schlagworte:	Algebra Algorithms Approximation B spline functions Computer Science Eigenvalues Nodes Numeric Computing Numerical Analysis Original Paper Polynomials Quadratures Theory of Computation Weighting functions 41A55 65D30 Moment Recurrence relation Gaussian quadrature formula Cardinal 65D32 spline Symbolic computation 41A15 Orthogonal polynomial 42C05
ISSN:	1017-1398, 1572-9265
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Abstract	The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B -splines φ m as the weight functions on [0, m ] ( m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.
AbstractList	The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B -splines φ m as the weight functions on [0, m ] ( m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated. The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φm as the weight functions on [0, m] (m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated.
Author	Milovanović, Gradimir V.
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Cites_doi	10.1137/1.9780898719727 10.1016/0771-050X(81)90008-5 10.1007/978-4-431-54919-2 10.1016/j.amc.2015.06.006 10.2307/2005948 10.1016/S0021-9800(69)80045-1 10.1090/S0025-5718-69-99647-1 10.1016/j.apnum.2014.11.010 10.1093/oso/9780198506720.001.0001 10.1145/174603.174605 10.1007/978-1-4614-7049-6_1 10.1007/978-3-540-68349-0 10.1007/s10543-008-0168-x 10.1137/1.9781611974300 10.1090/S0025-5718-06-01855-2 10.1137/0903018 10.1007/BF02437506 10.1006/acha.1999.0306 10.1016/j.aml.2010.06.029
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Comp.19692322123024520110.1090/S0025-5718-69-99647-10179.21901 – reference: ArakawaTIbukiyamaTKanekoMBernoulli Numbers and Zeta Functions. Springer Monographs in Mathematics2014TokyoSpringer1312.11015 – reference: CalabròFCorbo EspositoAManticaGRadiceTRefinable functions, functionals, and iterated function systemsAppl. Math. Comput.20162721992073418124 – reference: ManticaGA stable Stieltjes technique for computing orthogonal polynomials and Jacobi matrices associated with a class of singular measuresConstr. Approx.199612509530141219710.1007/BF024375060878.42014 – reference: PhillipsJLHansonRJGauss quadrature rules with B-spline weight functionsMath. Comp.197418Rewiew 2866634355110.2307/2005948[Loose microfiche suppl. A1–C4] – reference: GautschiWGoriLPitolliFGauss quadrature for refinable weight functionsAppl. Comput. Harmon. Anal.200083249257175492610.1006/acha.1999.03060954.65018 – reference: NeumanEMoments and Fourier transform of B-splinesJ. Comput. Appl. Math.19817516261195110.1016/0771-050X(81)90008-50452.42006 – reference: CalabròFManniCPitolliFComputation of quadrature rules for integration with respect to refinable functions on assigned nodesAppl. Numer. Math.201590168189330090110.1016/j.apnum.2014.11.0101326.65032 – reference: GautschiWOrthogonal Polynomials: Computation and Approximation2004OxfordClarendon Press1130.42300 – reference: GautschiWOrthogonal Polynomials in Matlab: Exercises and Solutions, Software–Environments–Tools2016Philadelphia, PASIAM06589760 – reference: ChuiCWavelets: A Mathematical Tool for Signal Analysis1997PhiladelphiaSIAM10.1137/1.97808987197270903.94007 – reference: Feller, W.: An Introduction to Probability Theory and Its Applications, vol. I. Wiley, New York (1950) – volume-title: Wavelets: A Mathematical Tool for Signal Analysis year: 1997 ident: 256_CR6 doi: 10.1137/1.9780898719727 – volume: 7 start-page: 51 year: 1981 ident: 256_CR21 publication-title: J. Comput. Appl. 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Math. doi: 10.1007/s10543-008-0168-x – volume-title: Orthogonal Polynomials in Matlab: Exercises and Solutions, Software–Environments–Tools year: 2016 ident: 256_CR13 doi: 10.1137/1.9781611974300 – volume: 75 start-page: 1891 year: 2006 ident: 256_CR14 publication-title: Math. Comput. doi: 10.1090/S0025-5718-06-01855-2 – volume: 3 start-page: 289 year: 1982 ident: 256_CR10 publication-title: SIAM J. Sci. Statist. Comput. doi: 10.1137/0903018 – volume: 12 start-page: 509 year: 1996 ident: 256_CR16 publication-title: Constr. Approx. doi: 10.1007/BF02437506 – volume: 8 start-page: 249 issue: 3 year: 2000 ident: 256_CR9 publication-title: Appl. Comput. Harmon. Anal. doi: 10.1006/acha.1999.0306 – volume: 23 start-page: 1346 year: 2010 ident: 256_CR20 publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2010.06.029
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Snippet	The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B -splines φ m as the weight... The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φm as the weight functions...
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SubjectTerms	Algebra Algorithms Approximation B spline functions Computer Science Eigenvalues Nodes Numeric Computing Numerical Analysis Original Paper Polynomials Quadratures Theory of Computation Weighting functions
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Title	Symbolic–numeric computation of orthogonal polynomials and Gaussian quadratures with respect to the cardinal B-spline
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