From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex
SIAM Journal on Numerical Analysis
Properties of multivariate homogeneous orthogonal polynomials
Journal of Approximation Theory
On polynomial interpolation of two variables
Journal of Approximation Theory
Improved Lebesgue constants on the triangle
Journal of Computational Physics
On the Lebesgue constant for the Xu interpolation formula
Journal of Approximation Theory
Bivariate Lagrange interpolation at the Padua points: The generating curve approach
Journal of Approximation Theory
Symbolic-numeric Gaussian cubature rules
Applied Numerical Mathematics
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In polynomial interpolation, the choice of the polynomial basis and the location of the interpolation points play an important role numerically, even more so in the multivariate case. We explore the concept of spherical orthogonality for multivariate polynomials in more detail on the disk. We focus on two items: on the one hand the construction of a fully orthogonal cartesian basis for the space of multivariate polynomials starting from this sequence of spherical orthogonal polynomials, and on the other hand the connection between these orthogonal polynomials and the Lebesgue constant in multivariate polynomial interpolation on the disk. We point out the many links of the two topics under discussion with the existing literature. The new results are illustrated with an example of polynomial interpolation and approximation on the unit disk. The numerical example is also compared with the popular radial basis function interpolation.