In this paper, we study the sequence of orthogonal polynomials ${\left\{S_n\right\}}_{n=0}^{\infty }$ with respect to the Sobolev-type inner product $〈f,g〉={\int }_{-1}^{1}f\left(x\right)g\left(x\right)d\mu \left(x\right)+\sum _{j=1}^N{\eta }_j{f}^{\left(d_j\right)}\left(c_j\right)g^{\left(d_j\right)}\left(c_j\right)$where $\mu$ is a finite positive Borel measure whose support $\mathrm{supp}\left(\mu \right)\subset [-1,1]$ contains an infinite set of points, ${\eta }_j>0$, $N,d_j\in {\mathbb{Z}}_{+}$ 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 ${\left\{S_n^{\left[k\right]}\right\}}_{n=0}^{\infty }$ are defined for each $k\in {\mathbb{Z}}_{+}$. If $\mu$ is in the Nevai class $\mathbf{M}(0,1)$, we prove an analogue of Markov’s Theorem on rational approximation to Markov type functions and prove that convergence takes place with geometric speed.

在本文中，我们研究正交多项式的序列 ${\left\{{\mathrm{小号}}_{ñ}\right\}}_{ñ=0}^{\infty }$ 关于Sobolev型内部产品 $〈F，G〉={\int }_{-\mathrm{1个}}^{\mathrm{1个}}F（X）G（X）d\mu （X）+\sum _{Ĵ=\mathrm{1个}}^{ñ}{\eta }_{Ĵ}{F}^{（d_{Ĵ}）}（C_{Ĵ}）G^{（d_{Ĵ}）}（C_{Ĵ}）$哪里 $\mu$ 是有限的正Borel测度，其支持 $\mathrm{支持}\left(\mu \right)\subset [-\mathrm{1个}，\mathrm{1个}]$ 包含无限个点 ${\eta }_{Ĵ}>0$， $ñ，d_{Ĵ}\in {\mathbb{ž}}_{+}$ 和 $\{C_{\mathrm{1个}}，\dots ，C_{ñ}\}\subset \mathbb{[R}\setminus [-\mathrm{1个}，\mathrm{1个}]$。在顺序的某些限制下，$〈\cdot ，\cdot 〉$，我们证明对于足够大的 $ñ$ 的零 ${\mathrm{小号}}_{ñ}$ 是真实，简单的 $ñ-ñ$ 他们躺在 $（-\mathrm{1个}，\mathrm{1个}）$ 和每个质量点 $C_{Ĵ}$ “吸引”剩余的人之一 $ñ$ 零。

相关多项式的序列 ${\left\{{\mathrm{小号}}_{ñ}^{\left[ķ\right]}\right\}}_{ñ=0}^{\infty }$ 为每个定义 $ķ\in {\mathbb{ž}}_{+}$。如果$\mu$ 在内娃课上 $\mathbf{中号}（0，\mathrm{1个}）$，我们证明了马尔可夫定理在对马尔可夫类型函数的有理逼近上的类似物，并证明了收敛速度是随着几何速度发生的。