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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
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