Recent Developments in Nonlinear Hyperbolic Conservation Laws 251 Orthogonal polynomials in several variables potentially useful in pde

R |xα| dμ(x) < ∞ (α ∈ (Z≥0)) and the support of μ has nonempty interior. Let Pn consist of all polynomials p of degree ≤ n such that ∫ Rd pq dμ = 0 for all polynomials q of degree < n. Then Pn has the same dimension ( n+d−1 n ) as the space of homogeneous polynomials of degree n in d variables. Furthermore, the spaces Pn (n = 0, 1, 2, . . .) are mutually orthogonal in L(μ). We call {Pn}n=0 a system of orthogonal polynomials with respect to the measure μ. As a refinement of this notion we may choose an orthogonal basis {pα}α1+···+αd=n for each space Pn, and call the polynomials pα orthogonal polynomials. Of course, there are many ways to choose such orthogonal bases. A system {Pn} of orthogonal polynomials in d variables is called classical if there is a second order pdo L acting on the space of polynomials such that Pn is an eigenspace of L for a certain eigenvalue λn (n = 0, 1, 2, . . .). As a refinement there may be, apart from L = L1, d − 1 further pdo’s L2, . . . , Ld such that L1, L2, . . . , Ld commute, are self-adjoint with respect to μ, and have onedimensional joint eigenspaces. Then we have OP’s pα with Ljpα = λ (j) α pα. It was shown by Krall & Sheffer [8] and Kwon, Lee & Littlejohn [9] that there are five families of classical orthogonal polynomials in 2 variables, as follows: