Sobolev orthogonal polynomials on product domains

Abstract

Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$ \langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y) \,dx\, dy + \lambda f(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a weight function on $[a_i,b_i]$ for $i = 1, 2$, $\lambda > 0$, and $(c_1, c_2)$ is a fixed point. The main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions, in particular, for product Laguerre and product Gegenbauer weight functions, which serve as primary examples.