Radial orthogonality and Lebesgue constants on the disk

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.

[1]  Marco Vianello,et al.  On the Lebesgue constant for the Xu interpolation formula , 2006, J. Approx. Theory.

[2]  Marco Vianello,et al.  Bivariate polynomial interpolation on the square at new nodal sets , 2005, Appl. Math. Comput..

[3]  Borislav Bojanov,et al.  On polynomial interpolation of two variables , 2003, J. Approx. Theory.

[4]  Walter Gautschi,et al.  On inverses of Vandermonde and confluent Vandermonde matrices. II , 1963 .

[5]  Walter Gautschi,et al.  On inverses of Vandermonde and confluent Vandermonde matrices , 1962 .

[6]  T. Bloom,et al.  Polynomial Interpolation and Approximation in C^d , 2011, 1111.6418.

[7]  Jan S. Hesthaven,et al.  From Electrostatics to Almost Optimal Nodal Sets for Polynomial Interpolation in a Simplex , 1998 .

[8]  Annie A. M. Cuyt,et al.  Multivariate orthogonal polynomials, homogeneous Padé approximants and Gaussian cubature , 2004, Numerical Algorithms.

[9]  Marco Vianello,et al.  Bivariate Lagrange interpolation at the Padua points: The generating curve approach , 2006, J. Approx. Theory.

[10]  Annie A. M. Cuyt,et al.  Properties of Multivariate Homogeneous Orthogonal Polynomials , 2001, J. Approx. Theory.

[11]  Yuan Xu,et al.  Regular points for Lagrange interpolation on the unit disk , 2005, Numerical Algorithms.

[12]  Humberto Rocha,et al.  On the selection of the most adequate radial basis function , 2009 .

[13]  Wilhelm Heinrichs,et al.  Improved Lebesgue constants on the triangle , 2005 .

[14]  Yuan Xu.  Polynomial Interpolation on the Unit Sphere and on the Unit Ball , 2004, Adv. Comput. Math..

[15]  J. Szabados,et al.  Interpolation of Functions , 1990 .

[16]  Symbolic-numeric Gaussian cubature rules , 2011 .

[17]  Burkhard Sündermann.  On projection constants of polynomial spaces on the unit ball in several variables , 1984 .