Orthogonal polynomials and partial differential equations on the unit ball

Abstract

Abstract. 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 equationThe 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.

On Sobolev orthogonal polynomials

“…For k ∈ N, the equation D −k Y = λ n Y of (11.3) was studied in [93], where a complete system of polynomial solutions was determined explicitly. For k ≥ 2, however, it is not known if the solutions are Sobolev orthogonal polynomials.…”

New steps on Sobolev orthogonality in two variables

“…We should mention here that this Sobolev inner product is a particular case of (10), which was previously studied in [7,5].…”

“…. is studied in [5]. Explicit polynomial solutions are constructed and the equation for µ = −2, −3, .…”

“…However, as far as we know, Sobolev orthogonal polynomials in several variables have been studied only in a very few particular cases. At the time of writing this paper, the only references in the subject are [4][5][6][7].…”

“…The three last references [5][6][7] are related to orthogonal polynomials on the unit ball B d := {x : ‖x‖ ≤ 1} of the Euclidean space R d , d ≥ 1.…”