This article is a first attempt to obtain weak limit formulas for weighted means of orthogonal polynomials. For this, we introduce a new mean Nevai class that guarantees the existence of a limiting measure for the weighted means. We show that for a family of measures in this mean Nevai class also the means of the Christoffel–Darboux kernels and the asymptotic distribution of the roots converge weakly to the same limiting measure. As a main example, we study the mean Nevai classes in which the limiting measure is the orthogonality measure of the ultraspherical polynomials. The respective weak limit formula can be regarded as an asymptotic weak addition formula for the corresponding class of measures.

中文翻译：

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正交多项式加权均值的弱极限

本文是首次尝试获得正交多项式加权均值的弱极限公式。为此，我们引入了一个新的均值Nevai类，该类保证了加权均值的限制度量的存在。我们表明，对于这个平均Nevai类中的一组度量，Christoffel–Darboux核的均值和根的渐近分布都微弱地收敛到同一限制度量。作为一个主要的例子，我们研究平均Nevai类，其中极限测度是超球形多项式的正交性测度。相应的弱极限公式可以看作是对应的一类措施的渐近弱加法公式。