Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Orthogonal polynomials in and on a quadratic surface of revolution

Authors: Sheehan Olver and Yuan Xu
Journal: Math. Comp. 89 (2020), 2847-2865
MSC (2010): Primary 42C05, 42C10, 65D15, 65D32
Published electronically: June 5, 2020
MathSciNet review: 4136549
Full-text PDF
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing spherical harmonics to the surface of a cone, hyperboloid, and paraboloid. We use this construction to develop cubature and fast approximation methods.

References [Enhancements On Off] (What's this?)

  • S. A. Agahanov, A method of constructing orthogonal polynomials of two variables for a certain class of weight functions, Vestnik Leningrad. Univ. 20 (1965), no. 19, 5–10 (Russian, with English summary). MR 0186841
  • Feng Dai and Yuan Xu, Approximation theory and harmonic analysis on spheres and balls, Springer Monographs in Mathematics, Springer, New York, 2013. MR 3060033
  • Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR 3289583
  • Tom Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 435–495. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. MR 0402146
  • Sheehan Olver and Yuan Xu, Orthogonal structure on a wedge and on the boundary of a square, Found. Comput. Math. 19 (2019), no. 3, 561–589. MR 3958801, DOI
  • S. Olver and Y. Xu, Orthogonal structure on a quadratic curve, IMA J. Numer. Anal., posted on April 18, 2020. DOI 10.1093/imanum/draa001 (to appear in print).
  • R. M. Slevinsky, Conquering the pre-computation in two-dimensional harmonic polynomial transforms, arXiv:1711.07866.
  • Richard Mikaël Slevinsky, On the use of Hahn’s asymptotic formula and stabilized recurrence for a fast, simple and stable Chebyshev-Jacobi transform, IMA J. Numer. Anal. 38 (2018), no. 1, 102–124. MR 3800016, DOI
  • Richard Mikaël Slevinsky, Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series, Appl. Comput. Harmon. Anal. 47 (2019), no. 3, 585–606. MR 3994987, DOI
  • B. Snowball and S. Olver, Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums, Stud. Appl. Maths., to appear. arXiv:1906.07962
  • Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR 0372517
  • G. Taylor, Disintegration of water droplets in an electric field, Proc. Roy. Soc. A, 280 (1964) 383–397.
  • Alex Townsend, Marcus Webb, and Sheehan Olver, Fast polynomial transforms based on Toeplitz and Hankel matrices, Math. Comp. 87 (2018), no. 312, 1913–1934. MR 3787396, DOI
  • Yuan Xu, Orthogonal polynomials and Fourier orthogonal series on a cone, J. Fourier Anal. Appl. 26 (2020), no. 3, Paper No. 36, 42. MR 4085344, DOI

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 42C05, 42C10, 65D15, 65D32

Retrieve articles in all journals with MSC (2010): 42C05, 42C10, 65D15, 65D32

Additional Information

Sheehan Olver
Affiliation: Department of Mathematics, Imperial College, London, United Kingdom
MR Author ID: 783322
ORCID: 0000-0001-6920-0826

Yuan Xu
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.
MR Author ID: 227532

Keywords: Orthogonal polynomials, quadratic surface of revolution, cubature rule, approximation method
Received by editor(s): June 25, 2019
Received by editor(s) in revised form: February 4, 2020
Published electronically: June 5, 2020
Additional Notes: The first author was supported by the Leverhulme Trust Research Project Grant RPG-2019-144 “Constructive approximation theory on and inside algebraic curves and surfaces”.
This work was supported by EPSRC grant no EP/K032208/1.
Article copyright: © Copyright 2020 American Mathematical Society