an:00770285
Zbl 0828.42014
Bos, Len
Asymptotics for the Christoffel function for Jacobi like weights on a ball in \(\mathbb{R}^ m\)
EN
N. Z. J. Math. 23, No. 2, 99-109 (1994).
1171-6096 1179-4984
1994
j
42C05 41A63
multivariate orthogonal polynomials; Jacobi like weights on the \(m\)- dimensional unit ball; Christoffel function
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.
M.G.de Bruin (Delft)