Abstract
The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function rho (t) such that rho (t)^2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.
Highlights
Krall and Sheffer classical bivariate polynomials can be constructed in this way
Where {p(nm)(t)}n≥0 is an orthogonal polynomial sequence (OPS in short) associated with the weight function ρ2m+1(t)ω1(t), m ≥ 0, and {qn(t)}n≥0 is an OPS associated with ω2(t)
Our most interesting result is the existence of connection formulas relating the bivariate Sobolev and standard orthogonal polynomials
Summary
If u, pn pm = 0, n = m, and u, p2n = hn = 0, n ≥ 0, we say that {pn(t)}n≥0 is an orthogonal polynomial sequence (OPS) associated with u. If an OPS associated with u exists, u is called quasi-definite. A moment functional u is positive definite if u, p2 > 0 for all non zero polynomial p ∈ Π; positive definite moment functionals are quasi-definite, and OPS associated with u exists. The product rule holds, that is, D(q u) = q u + q Du. A quasi-definite moment functional u is called classical (see, for instance, [10]) if there exist non-zero polynomials φ(t) and ψ(t) with deg φ ≤ 2 and deg ψ = 1, such that u satisfies the distributional Pearson equation.
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