Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Sobolev orthogonal polynomials of d variables on the product domain Ω : = [ a 1 , b 1 ] × ⋯ × [ a d , b d ] with respect to the inner product f , g S = c ∫ Ω ∇ κ f ( x ) · ∇ κ g ( x ) W ( x ) d x + ∑ i = 0 κ - 1 λ i ∇ i f ( p ) · ∇ i g ( p ) , κ ∈ N , are constructed, where ∇ i f , i = 0 , 1 , 2 , … , κ , is a column vector which contains all the partial derivatives of order i of f , x : = ( x 1 , x 2 , … , x d ) ∈ R d , d x : = d x 1 d x 2 ⋯ d x d , W ( x ) : = w 1 ( x 1 ) w 2 ( x 2 ) ⋯ w d ( x d ) is a product weight function on Ω , w i is a weight function on [ a i , b i ] , i = 1 , 2 , … , d , λ i > 0 for i = 0 , 1 , … , κ - 1 , p = ( p 1 , p 2 , … , p d ) is a point in R d , typically on the boundary of Ω , and c is the normalization constant of W . The main result consists of a generalization to several variables and higher order derivatives of some results which are presented in the literature of Sobolev orthogonal polynomials in two variables; namely, properties involving the integral part in · , · S , a connection formula, and a recursive relation for constructing iteratively the polynomials. To illustrate the main ideas, we present a new example for the Hermite–Hermite–Laguerre product weight function.