Sobolev Orthogonal Polynomials of Several Variables on Product Domains

Sobolev orthogonal polynomials of d variables on the product domain Ω:=[a1,b1]×⋯×[ad,bd]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega :=[a_1,b_1]\times \cdots \times [a_d,b_d]$$\end{document} with respect to the inner product f,gS=c∫Ω∇κf(x)·∇κg(x)W(x)dx+∑i=0κ-1λi∇if(p)·∇ig(p),κ∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\langle f,g\right\rangle _S= c\int _\Omega \nabla ^\kappa f({\mathbf {x}})\cdot \nabla ^\kappa g({\mathbf {x}})W({\mathbf {x}}){\mathrm{d}}{\mathbf {x}}+ \sum _{i=0}^{\kappa -1}\lambda _i \nabla ^ i f({\mathbf {p}})\cdot \nabla ^i g({\mathbf {p}}), \kappa \in {\mathbb {N}}, \end{aligned}$$\end{document}are constructed, where ∇if\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^i f$$\end{document}, i=0,1,2,…,κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=0,1,2,\ldots ,\kappa $$\end{document}, is a column vector which contains all the partial derivatives of order i of f, x:=(x1,x2,…,xd)∈Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {x}}:=(x_1,x_2,\ldots ,x_d)\in {\mathbb {R}}^d$$\end{document}, dx:=dx1dx2⋯dxd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{d}}{\mathbf {x}}:={\mathrm{d}}x_1{\mathrm{d}}x_2\cdots {\mathrm{d}}x_d$$\end{document}, W(x):=w1(x1)w2(x2)⋯wd(xd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W({\mathbf {x}}):=w_1(x_1)w_2(x_2)\cdots w_d(x_d)$$\end{document} is a product weight function on Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, wi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_i$$\end{document} is a weight function on [ai,bi]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[a_i,b_i]$$\end{document}, i=1,2,…,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,2,\ldots ,d$$\end{document}, λi>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _i >0$$\end{document} for i=0,1,…,κ-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=0,1,\ldots ,\kappa -1$$\end{document}, p=(p1,p2,…,pd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {p}}=(p_1,p_2,\ldots ,p_d)$$\end{document} is a point in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d$$\end{document}, typically on the boundary of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\langle \cdot ,\cdot \right\rangle _S$$\end{document}, 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.