Abstract
It is well-known that the classical univariate orthogonal polynomials give rise to highly efficient Gaussian quadrature rules. We show how the classical orthogonal polynomials can be generalized to a multivariate setting and how this generalization leads to Gaussian cubature rules for specific families of multivariate polynomials.
The multivariate homogeneous orthogonal functions that we discuss here satisfy a unique slice projection property: they project to univariate orthogonal polynomials on every one-dimensional subspace spanned by a vector from the unit hypersphere. We therefore call them spherical orthogonal polynomials.
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Benouahmane, B., Cuyt, A.: Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature. Numer. Algor. 24, 1–15 (2000)
Benouahmane, B., Cuyt, A.: Properties of multivariate homogeneous orthogonal polynomials. J. Approx. Theory 113, 1–20 (2001)
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© 2004 Springer-Verlag Berlin Heidelberg
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Cuyt, A., Benouahmane, B., Verdonk, B. (2004). Spherical Orthogonal Polynomials and Symbolic-Numeric Gaussian Cubature Formulas. In: Bubak, M., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds) Computational Science - ICCS 2004. ICCS 2004. Lecture Notes in Computer Science, vol 3037. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24687-9_71
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DOI: https://doi.org/10.1007/978-3-540-24687-9_71
Publisher Name: Springer, Berlin, Heidelberg
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