article

Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions

Authors:

Gradimir V. Milovanović,

Aleksandar S. CvetkovićAuthors Info & Claims

Journal of Computational and Applied Mathematics, Volume 179, Issue 1-2

Pages 263 - 287

Published: 01 July 2005 Publication History

Abstract

In this paper we consider polynomials orthogonal with respect to an oscillatory weight function w(x)=xe^i^m^@p^x on [-1,1], where m is an integer. The existence of such polynomials as well as several of their properties (three-term recurrence relation, differential equation, etc.) are proved. We also consider related quadrature rules and give applications of such quadrature rules to some classes of integrals involving highly oscillatory integrands.

References

[1]

B. Beckerman, M. Castro, On the determinacy of the complex Jacobi operators, Publication ANO 446, Université Lille, 2002.

[2]

Chihara, T.S., An Introduction to Orthogonal Polynomials. 1978. Gordon and Breach, New York.

[3]

A.S. Cvetković, G.V. Milovanović, The Mathematica Package "Orthogonal Polynomials", Facta Univ. Ser. Math. Inform 19 (2004).

[4]

de Bruin, M.G., Polynomials orthogonal on a circular arc. J. Comput. Appl. Math. v31. 253-266.

[5]

Gautschi, W., Algorithm 726: ORTHPOL-A package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. . ACM Trans. Math. Software. v10. 21-62.

[6]

Gautschi, W., Landau, H.J. and Milovanović, G.V., Polynomials orthogonal on the semicircle, II. Constr. Approx. v3. 389-404.

[7]

Gautschi, W. and Milovanović, G.V., Polynomials orthogonal on the semicircle. J. Approx. Theory. v46. 230-250.

[8]

Golub, G.H. and Welsch, J.H., Calculation of Gauss quadrature rule. Math. Comput. v23. 221-230.

[9]

Gonzales-Vera, P., Lopez, G., Orive, R. and Santos, J.C., On the convergence of quadrature formula for complex weight functions. J. Math. Anal. Appl. v189. 514-532.

[10]

Magnus, A.P., Toeplitz matrix techniques and convergence of complex weight Pade approximation. J. Comput. Appl. Math. v19. 23-28.

[11]

Magnus, A.P., Painlevé-type of differential equations for the recurrence coefficients of semi-classical orthogonal polynomials. J. Comput. Appl. Math. v57. 215-237.

Digital Library

[12]

Mate, A. and Nevai, P., A generalization of Poincare theorem for recurrence equations. J. Approx. Theory. v63. 92-97.

[13]

G.V. Milovanović, Complex orthogonality on the semicircle with respect to Gegenbauer weight: theory and applications, in: Th.M. Rassias (Ed.), Topics in Mathematical Analysis, Ser. Pure Math., 11, World Scientific Publishing, Teaneck, NJ, 1989, pp. 695-722.

[14]

Milovanović, G.V., Numerical Analysis, Part I. 1991. Naučna knjiga, Belgrade.

[15]

Milovanović, G.V., Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures. Comput. Math. Appl. v36. 19-39.

[16]

Milovanović, G.V. and Cvetković, A.S., Note on construction of weights in Gauss-type quadrature rule. Facta Univer. Ser. Math. Inform. v15. 69-83.

[17]

Milovanović, G.V. and Cvetković, A.S., Complex Jacobi matrices and quadrature rules. FILOMAT. v17. 117-134.

[18]

Milovanović, G.V. and Cvetković, A.S., An application of little 1/q-Jacobi polynomials to summation of certain series. Facta Univ. Ser. Math. Inform. v18. 31-46.

[19]

Milovanović, G.V. and Rajković, P.M., On polynomials orthogonal on a circular arc. J. Comput. Appl. Math. v51. 1-13.

Digital Library

[20]

Nuttal, J. and Wherry, C.J., Gaussian integration for complex weight functions. J. Inst. Maths. Appl. v21. 165-170.

[21]

Pasquini, L., Accurate computation of the zeroes of the generalized Bessel polynomials. Numer. Math. v86. 507-538.

[22]

Stahl, H., Spurious poles in Pade approximation. J. Comput. Appl. Math. v99. 511-527.

Cited By

Gözüirmizi CDemiralp M(2018)The application of the fluctuation expansion with extended basis set to numerical integrationWSEAS Transactions on Mathematics10.5555/1558830.15588338:5(205-212)Online publication date: 20-Dec-2018
https://dl.acm.org/doi/10.5555/1558830.1558833
Gözükirmizi CDemiralp M(2009)The application of the fluctuation expansion with extended basis set to numerical integrationProceedings of the 2nd WSEAS international conference on Multivariate analysis and its application in science and engineering10.5555/1561887.1561903(93-100)Online publication date: 30-May-2009
https://dl.acm.org/doi/10.5555/1561887.1561903

Index Terms

Orthogonal polynomials and Gaussian quadrature rules related to oscillatory weight functions
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Generating functions
  2. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials
    2. Quadrature

Recommendations

Gaussian type quadrature rules related to the oscillatory modification of the generalized Laguerre weight functions
Abstract
In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions w α ( x ) = x α e − x, α > − 1, on ( 0 , + ∞ ) by the oscillatory factor exp ( i ζ x ), where ζ > 0. For calculation of such ...
Generalized quadrature rules of Gaussian type for numerical evaluation of singular integrals

An efficient method for constructing a class of generalized quadrature formulae of Gaussian type on ( - 1 , 1 ) for integrands having logarithmic singularities is developed. That kind of singular integrals are very common in the boundary element method. ...
Asymptotic properties of Laguerre---Sobolev type orthogonal polynomials

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product $$ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\...

Comments

Information & Contributors

Information

Published In

cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 179, Issue 1-2

Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad

1 July 2005

441 pages

ISSN:0377-0427

Issue’s Table of Contents

Copyright © Elsevier B.V. © 2004.

Publisher

Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 01 July 2005

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

2
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 19 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Gözüirmizi CDemiralp M(2018)The application of the fluctuation expansion with extended basis set to numerical integrationWSEAS Transactions on Mathematics10.5555/1558830.15588338:5(205-212)Online publication date: 20-Dec-2018
https://dl.acm.org/doi/10.5555/1558830.1558833
Gözükirmizi CDemiralp M(2009)The application of the fluctuation expansion with extended basis set to numerical integrationProceedings of the 2nd WSEAS international conference on Multivariate analysis and its application in science and engineering10.5555/1561887.1561903(93-100)Online publication date: 30-May-2009
https://dl.acm.org/doi/10.5555/1561887.1561903

View Options

View options

Figures

Tables

Media

View Issue’s Table of Contents