## Orthogonal polynomials in and on a quadratic surface of revolution

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Sheehan Olver and Yuan Xu
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## 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

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## Additional Information

**Sheehan Olver**- Affiliation: Department of Mathematics, Imperial College, London, United Kingdom
- MR Author ID: 783322
- ORCID: 0000-0001-6920-0826
- Email: s.olver@imperial.ac.uk
**Yuan Xu**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222.
- MR Author ID: 227532
- Email: yuan@uoregon.edu
- 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