On Sobolev type orthogonality

Abstract

We generalize the Skrzipek``s methods in the case of Sobolev type inner products and consider the following problem : Generate a sequence $\{Q_n\}$ of polynomials, $deg(Q_n)=n$, orthogonal with respect to inner product defined by $$(f,g)=\int_I fg\, d\mu+ \sum_{p,q=1}^K\sum_{i=0}^{n_p-1}\sum_{j=0}^{n_q-1} \lambda_{p,q}^{i,j} f^{(i)}(c_p)g^{(j)}(c_q), $$ where $d\mu$ is a positive measure on an interval I, $n_p$, $1\le p\le K$ are nonnegative intergers, $c_p\in R$ and $\lambda_{p,q}^{i,j}= \lambda_{q,p}^{j,i}\ge0$. Next, We are concerned with the representation formula and behavior of zeros of Sobolev orthogonal polynomials which are orthogonal relative to a Sobolev pseudo-inner product of type $$ \phi (p,q) := \int_I p(x)q(x)\, d\sigma (x) + \int_{I^{\prime}} p^{\prime}(x) q^{\prime}(x)\, d\mu (x),$$ where $d\sigma$ and $d\mu\, (\ne 0)$ are Borel measures on intervals I and $I^{\prime}$ respectively.