The condition of Vandermonde-like matrices involving orthogonal polynomials

To my teacher, Alexander M. Ostrowski, in gratitude on his 90th birthday
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Abstract

The condition number (relative to the Frobenius norm) of the n × n matrix Pn = [pi−1(xj)]i, j = 1n is investigated, where pr(·) = pr(·; ) are orthogonal polynomials with respect to some weight distribution and xj are pairwise distinct real numbers. If the nodes xj are the zeros of pn, the condition number is either expressed, or estimated from below and above, in terms of the Christoffel numbers for , depending on whether the pr are normalized or not. For arbitrary real xj and normalized pr a lower bound of the condition number is obtained in terms of the Christoffel function evaluated at the nodes. Numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions .

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Work supported in part by the National Science Foundation under grant MCS-7927158.