论文信息 - Orthogonal polynomials and partial differential equations on the unit ball

Orthogonal polynomials and partial differential equations on the unit ball

Orthogonal polynomials of degree n with respect to the weight function W μ (x) = (1 - ∥x∥ 2 ) μ on the unit ball in R are known to satisfy the partial differential equation [Δ- ?x, ∇? 2 - (2μ + d) ?x, ∇?] P = -n(n + 2μ + d)P for μ > -1. The singular case of μ = -1, -2, ... is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = -2, -3, ... is shown to have complete polynomial solutions if the dimension d is odd. The orthogonality of the solution is also discussed.

Yuan Xu | Miguel Pinar

[1] K. H. Kwon,et al. SOBOLEV ORTHOGONAL POLYNOMIALS RELATIVE TO ${\lambda}$p(c)q(c) + , 1997 .

[2] G. Szegő. Zeros of orthogonal polynomials , 1939 .

[3] Yuan Xu,et al. Orthogonal Polynomials of Several Variables: Subject index , 2001 .

[4] Sobolev orthogonal polynomials: The discrete-continuous case , 1999 .

[5] H. L. Krall,et al. Orthogonal polynomials in two variables , 1967 .

[6] Yuan Xu,et al. A family of Sobolev orthogonal polynomials on the unit ball , 2005, J. Approx. Theory.

[7] Lance L. Littlejohn,et al. Sobolev orthogonal polynomials in two variables and second order partial differential equations , 2006 .

[8] K. Kwon,et al. Orthogonal polynomials in two variables and second-order partial differential equations , 1997 .

[9] Orthogonality of the Jacobi polynomials with negative integer parameters , 2002 .

[10] Yuan Xu. Sobolev orthogonal polynomials defined via gradient on the unit ball , 2008, J. Approx. Theory.

[11] Sobolev orthogonality for the Gegenbauer polynomials {C n (-N+ 1 2 ) } n?0 , 1998 .

[12] Jasper V. Stokman,et al. Orthogonal Polynomials of Several Variables , 2001, J. Approx. Theory.

[13] Kendall E. Atkinson,et al. Solving the Nonlinear Poisson Equation on the Unit Disk , 2005 .