Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Abstract

Sobolev orthogonal polynomials of d variables on the product domain $\Omega :=[a_1,b_1]\times \cdots \times [a_d,b_d]$ with respect to the inner product

$$\begin{aligned} \left\langle f,g\right\rangle _S= c\int _\Omega \nabla ^\kappa f({\mathbf {x}})\cdot \nabla ^\kappa g({\mathbf {x}})W({\mathbf {x}}){\mathrm{d}}{\mathbf {x}}+ \sum _{i=0}^{\kappa -1}\lambda _i \nabla ^ i f({\mathbf {p}})\cdot \nabla ^i g({\mathbf {p}}), \kappa \in {\mathbb {N}}, \end{aligned}$$

are constructed, where $\nabla ^i f$, $i=0,1,2,\ldots ,\kappa$, is a column vector which contains all the partial derivatives of order i of f, ${\mathbf {x}}:=(x_1,x_2,\ldots ,x_d)\in {\mathbb {R}}^d$, ${\mathrm{d}}{\mathbf {x}}:={\mathrm{d}}x_1{\mathrm{d}}x_2\cdots {\mathrm{d}}x_d$, $W({\mathbf {x}}):=w_1(x_1)w_2(x_2)\cdots w_d(x_d)$ is a product weight function on $\Omega$, $w_i$ is a weight function on $[a_i,b_i]$, $i=1,2,\ldots ,d$, $\lambda _i >0$ for $i=0,1,\ldots ,\kappa -1$, ${\mathbf {p}}=(p_1,p_2,\ldots ,p_d)$ is a point in ${\mathbb {R}}^d$, typically on the boundary of $\Omega$, 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 $\left\langle \cdot ,\cdot \right\rangle _S$, 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.