For every system $\left\{p_n\right(z){\}}_{n=0}^{\mathrm{\infty }}$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n\left(z\right)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending on an arbitrary number of complex parameters) which are generalized eigenvalues of a difference operator (in n) and generalized eigenvalues of a differential operator (in z) are given. We define suitable Sobolev spaces with matrix weights and consider measurable factorizations of weights. Applications of a general connection between Sobolev orthogonal polynomials and orthogonal systems of functions in the direct sum of scalar $L_{\mu }^2$ spaces are discussed.

对于每个系统 $\{p_{ñ}（ž）{\}}_{ñ=0}^{\mathrm{\infty }}$ 关于OPRL或OPUC，我们构造Sobolev正交多项式 ${ÿ}_{ñ}（ž）$，其中涉及明确的整数表示 $p_{ñ}$。给出了两个具体的Sobolev正交多项式族（取决于任意数量的复杂参数），它们是差分算子的广义特征值（在n中）和微分算子的广义特征值（在z中）。我们用矩阵权重定义合适的Sobolev空间，并考虑权重的可测量因式分解。Sobolev正交多项式与函数正交系统之间的一般联系在标量直接和中的应用${\mathrm{大号}}_{\mu }^{\mathrm{2个}}$ 讨论了空格。