Volume 14, pp. 152-164, 2002.

Bounds for Vandermonde type determinants of orthogonal polynomials

Gerhard Schmeisser

Abstract

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