Orthogonal polynomials for refinable linear functionals
Authors:
Dirk Laurie and Johan de Villiers
Journal:
Math. Comp. 75 (2006), 1891-1903
MSC (2000):
Primary 65D30, 42C40; Secondary 42C05, 65D07
DOI:
https://doi.org/10.1090/S0025-5718-06-01855-2
Published electronically:
May 23, 2006
MathSciNet review:
2240640
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: A refinable linear functional is one that can be expressed as a convex combination and defined by a finite number of mask coefficients of certain stretched and shifted replicas of itself. The notion generalizes an integral weighted by a refinable function. The key to calculating a Gaussian quadrature formula for such a functional is to find the three-term recursion coefficients for the polynomials orthogonal with respect to that functional. We show how to obtain the recursion coefficients by using only the mask coefficients, and without the aid of modified moments. Our result implies the existence of the corresponding refinable functional whenever the mask coefficients are nonnegative, even when the same mask does not define a refinable function. The algorithm requires rational operations and, thus, can in principle deliver exact results. Numerical evidence suggests that it is also effective in floating-point arithmetic.
- 1. A. Barinka, T. Barsch, S. Dahlke, and M. Konik, Some remarks on quadrature formulas for refinable functions and wavelets, ZAMM Z. Angew. Math. Mech. 81 (2001), no. 12, 839–855 (English, with English and German summaries). MR 1872770, https://doi.org/10.1002/1521-4001(200112)81:12<839::AID-ZAMM839>3.0.CO;2-F
- 2. Arne Barinka, Titus Barsch, Stephan Dahlke, Mario Mommer, and Michael Konik, Quadrature formulas for refinable functions and wavelets. II. Error analysis, J. Comput. Anal. Appl. 4 (2002), no. 4, 339–361. MR 1933926, https://doi.org/10.1023/A:1019959727304
- 3. Bernhard Beckermann and Emmanuel Bourreau, How to choose modified moments?, J. Comput. Appl. Math. 98 (1998), no. 1, 81–98. MR 1656990, https://doi.org/10.1016/S0377-0427(98)00116-2
- 4. T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR 0481884
- 5. Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
- 6. Walter Gautschi, Questions of numerical condition related to polynomials, Studies in numerical analysis, MAA Stud. Math., vol. 24, Math. Assoc. America, Washington, DC, 1984, pp. 140–177. MR 925213
- 7. Walter Gautschi, Orthogonal polynomials: computation and approximation, Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2004. Oxford Science Publications. MR 2061539
- 8. Walter Gautschi, Laura Gori, and Francesca Pitolli, Gauss quadrature for refinable weight functions, Appl. Comput. Harmon. Anal. 8 (2000), no. 3, 249–257. MR 1754926, https://doi.org/10.1006/acha.1999.0306
- 9. Walter Gautschi, Laura Gori, and Francesca Pitolli, Gauss quadrature for refinable weight functions, Appl. Comput. Harmon. Anal. 8 (2000), no. 3, 249–257. MR 1754926, https://doi.org/10.1006/acha.1999.0306
- 10. Daan Huybrechs and Stefan Vandewalle, Composite quadrature formulae for the approximation of wavelet coefficients of piecewise smooth and singular functions, J. Comput. Appl. Math. 180 (2005), no. 1, 119–135. MR 2141488, https://doi.org/10.1016/j.cam.2004.10.005
- 11. D. P. Laurie and J. M. de Villiers, Orthogonal polynomials and Gaussian quadrature for refinable weight functions, Appl. Comput. Harmon. Anal. 17 (2004), no. 3, 241–258. MR 2097078, https://doi.org/10.1016/j.acha.2004.06.002
- 12. Charles A. Micchelli, Mathematical aspects of geometric modeling, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 65, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR 1308048
- 13.
Pari-GP.
URL=http://pari.math.u-bordeaux.fr. An interactive programming environment for doing formal computations on recursive types, including rational and multiprecision floating-point numbers, polynomials and truncated power series. - 14. James L. Phillips and Richard J. Hanson, Gauss quadrature rules with 𝐵-spline weight functions, Math. Comp. 28 (1974), no. 126, loose microfiche suppl, A1–C4. MR 343551, https://doi.org/10.2307/2005948
- 15. R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math. 18 (1971/72), 465–478. MR 303693, https://doi.org/10.1007/BF01406683
- 16. A. H. Stroud and Don Secrest, Gaussian quadrature formulas, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966. MR 0202312
- 17. Wim Sweldens and Robert Piessens, Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions, SIAM J. Numer. Anal. 31 (1994), no. 4, 1240–1264. MR 1286226, https://doi.org/10.1137/0731065
Retrieve articles in Mathematics of Computation with MSC (2000): 65D30, 42C40, 42C05, 65D07
Retrieve articles in all journals with MSC (2000): 65D30, 42C40, 42C05, 65D07
Additional Information
Dirk Laurie
Affiliation:
Department of Mathematics, University of Stellenbosch, South Africa
Email:
dpl@sun.ac.za
Johan de Villiers
Affiliation:
Department of Mathematics, University of Stellenbosch, South Africa
Email:
jmdv@sun.ac.za
DOI:
https://doi.org/10.1090/S0025-5718-06-01855-2
Keywords:
Gaussian quadrature,
refinable,
orthogonal polynomials
Received by editor(s):
October 22, 2004
Received by editor(s) in revised form:
May 3, 2005
Published electronically:
May 23, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.