Sobolev orthogonal polynomials of high order in two variables defined on product domains

ABSTRACT

Let f and g real functions of two variables and ${\mathrm{\nabla }}^{2}f(x,y)=[f_{xx},f_{xy},f_{yx},f_{yy}].$ We consider the polynomials orthogonal with respect to the inner product: $⟨f,g{⟩}_S=c{\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}^{2}f(x,y)\cdot {\mathrm{\nabla }}^{2}g(x,y)W(x,y)\mathrm{d}y\mathrm{d}x+\lambda f(c_1,c_2)g(c_1,c_2),$ where $(c_1,c_2)$ is some corner point in $\mathrm{\Omega }=[a_1,b_1]\times [a_2,b_2]$, $\lambda >0$, and $c=1/{\int }_{\mathrm{\Omega }}W(x,y)\mathrm{d}x\mathrm{d}y$. We study the particular cases when $W(x,y)$ correspond to the product of the weight of Laguerre polynomials and Gegenbauer polynomials.

KEYWORDS:

• Orthogonal polynomials
• several variables
• classical orthogonal polynomials
• product domains
• Sobolev polynomials

AMS CLASSIFICATION:

• 33C50
• 42C10

Acknowledgements

The authors thank the rigorous and valuable comments from the referees. They greatly contributed to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.