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 $$⟨f,g⟩={\int }_{-1}^{1}f(x)g(x)d\mu (x)+\sum _{j=1}^N{\eta }_j{f}^{(d_j)}(c_j)g^{(d_j)}(c_j),$$ 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 $⟨\cdot ,\cdot ⟩$, 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.