Mediterranean Journal of Mathematics ( IF 1.1 ) Pub Date : 2021-09-14 , DOI: 10.1007/s00009-021-01852-z Herbert Dueñas Ruiz 1 , Omar Salazar-Morales 1 , Miguel Piñar 2
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.
中文翻译：
乘积域上多个变量的 Sobolev 正交多项式
乘积域上d 个变量的Sobolev 正交多项式\(\Omega :=[a_1,b_1]\times \cdots \times [a_d,b_d]\)关于内积
$$\begin{aligned} \left\langle f,g\right\rangle _S= c\int _\Omega \nabla ^\kappa f({\mathbf {x}})\cdot \nabla ^\kappa g( {\mathbf {x}})W({\mathbf {x}}){\mathrm{d}}{\mathbf {x}}+ \sum _{i=0}^{\kappa -1}\lambda _i \nabla ^ if({\mathbf {p}})\cdot \nabla ^ig({\mathbf {p}}), \kappa \in {\mathbb {N}}, \end{aligned}$$被构造，其中\(\nabla ^if\) , \(i=0,1,2,\ldots ,\kappa \)是一个列向量，其中包含f阶i的所有偏导数，\({ \mathbf {x}}:=(x_1,x_2,\ldots ,x_d)\in {\mathbb {R}}^d\) , \({\mathrm{d}}{\mathbf {x}}:= {\mathrm{d}}x_1{\mathrm{d}}x_2\cdots {\mathrm{d}}x_d\) , \(W({\mathbf {x}}):=w_1(x_1)w_2(x_2 )\cdots w_d(x_d)\)是\(\Omega \)上的乘积权重函数，\(w_i\)是\([a_i,b_i]\)上的权重函数，\(i=1,2, \ldots ,d\) , \(\lambda _i >0\)为\(i=0,1,\ldots ,\kappa -1\), \({\mathbf {p}}=(p_1,p_2,\ldots ,p_d)\)是\({\mathbb {R}}^d\) 中的一个点，通常在\(\Omega \)，而c是W的归一化常数。主要结果包括对几个变量的泛化和一些结果的高阶导数，这些结果在两个变量的 Sobolev 正交多项式的文献中提出；即涉及\(\left\langle \cdot ,\cdot \right\rangle _S\)中的积分部分的性质、连接公式和用于迭代构造多项式的递归关系。为了说明主要思想，我们展示了 Hermite-Hermite-Laguerre 乘积权重函数的新示例。
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