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.
- 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, DOI 10.1007/978-1-4614-6660-4
- 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, DOI 10.1017/CBO9781107786134
- 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) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 435–495. 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 10.1007/s10208-018-9393-0
- 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 10.1093/imanum/drw070
- 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 10.1016/j.acha.2017.11.001
- 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 Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. 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 10.1090/mcom/3277
- 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 10.1007/s00041-020-09741-x
- Sheehan Olver
- Affiliation: Department of Mathematics, Imperial College, London, United Kingdom
- MR Author ID: 783322
- ORCID: 0000-0001-6920-0826
- Email: firstname.lastname@example.org
- Yuan Xu
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.
- MR Author ID: 227532
- Email: email@example.com
- 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.
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp. 89 (2020), 2847-2865
- MSC (2010): Primary 42C05, 42C10, 65D15, 65D32
- DOI: https://doi.org/10.1090/mcom/3544
- MathSciNet review: 4136549