Volume 14, pp. 152-164, 2002.

Bounds for Vandermonde type determinants of orthogonal polynomials

Gerhard Schmeisser

Abstract

Let $\,(P_n)_{n\in {\bf N}_0}$ be a system of monic orthogonal polynomials. We establish upper and lower estimates for determinants of the form $$V_n(z_1,\dots, z_k)\,:=\, {\rm det} \left( \begin{array}{ccc} P_n(z_1) & \dots & P_{n+k-1}(z_1)\\ \vdots & & \vdots\\ P_n(z_k) & \dots & P_{n+k-1}(z_k) \end{array} \right). $$ For the proofs, we have to study the monic orthogonal system $\,(P_n^{[w]})_{n\in {\bf N}_0}$ obtained by inserting the polynomial $w(x):=\prod_{\nu=1}^k(x-z_\nu)$ as a weight into the inner product defining $\,(P_n)_{n\in {\bf N}_0}$. We also express the recurrence formula for $\,(P_n^{[w]})_{n\in {\bf N}_0}$ in terms of Vandermonde type determinants.

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Key words

Vandermonde type determinants, orthogonal systems, polynomial weights, inequalities.

AMS subject classifications

42C05, 15A15, 15A45, 30A10.

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