Rational Approximation and Sobolev-type Orthogonality

Abstract

In this paper, we study the sequence of orthogonal polynomials $\{S_n\}_{n=0}^{\infty}$ with respect to the Sobolev-type inner product $$\langle f,g \rangle= \int_{-1}^{1} f(x) g(x) \,d\mu(x) +\sum_{j=1}^{N} \eta_{j} \,f^{(d_j)}(c_{j}) g^{(d_j)}(c_{j}), $$ where $\mu$ is in the Nevai class $\mathbf{M}(0,1)$, $\eta_j >0$, $N,d_j \in \mathbb{Z}_{+}$ and $\{c_1,\dots,c_N\}\subset \mathbb{R} \setminus [-1,1]$. Under some restriction of order in the discrete part of $\langle\cdot,\cdot \rangle$, we prove that for sufficiently large $n$ the zeros of $S_n$ are real, simple, $n-N$ of them lie on $(-1,1)$ and each of the mass points $c_j$ ``attracts'' one of the remaining $N$ zeros. The sequences of associated polynomials $\{S_n^{[k]}\}_{n=0}^{\infty}$ are defined for each $k\in \mathbb{Z}_{+}$. We prove an analogous of Markov's Theorem on rational approximation to a function of certain class of holomorphic functions and we give an estimate of the ``speed'' of convergence.