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Asymptotics for the Christoffel function for Jacobi like weights on a ball in $$\mathbb{R}^ m$$. (English) Zbl 0828.42014
Let $$K^{(\alpha)}_n(x, y)$$ be the Christoffel function for the polynomials orthogonal on the $$m$$-dimensional unit ball with respect to the weight function $$w(x)= (1- |x|^2)^\alpha$$, $$\alpha\geq - 1/2$$.
The main result of the paper is: $\lim_{n\to \infty} {n+m\choose m}^{-1} K^{(\alpha)}_n(x, x)= {2\over \omega_m} {1\over \sqrt{1- |x |^2}} {1\over w(x)},$ where $$\omega_m$$ is the area of the unit sphere in $$\mathbb{R}^{m+ 1}$$. This extends known results in the univariate case and was proved in the multivariate case for $$\alpha= \pm{1\over 2}$$ only.
The proof follows after a chain of lemmas and is almost entirely self- contained. A nice piece of work.

##### MSC:
 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 41A63 Multidimensional problems