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On orthogonal polynomials with respect to a positive definite matrix of measures. (English) Zbl 0832.42014

The author states a generalization of the so-called Favard theorem, giving a connection between the existence of a \((2N + 1)\)-term recurrence relation for the polynomials \(p_n (t)\), \(\deg p_n = n\), of the form \[ h (t)p_n (t) = c_{n, 0} p_n (t) + \sum^N_{k = 1} \bigl( c_{n, k} p_{n - k} (t) + c_{n +k,k} p_{n + k} ( t ) \bigr) \] \((c_{n,k}\) real numbers, \(p_k (t) = 0\) for \(k < 0)\) with \(h(t)\) a real polynomial of degree \(N\) and the existence of a positive definite \(N \times N\) matrix of measures with respect to which the polynomials \(p_n\) are orthogonal. As well as in the original Favard version \((N = 1, h(t) = t)\) quoted in the paper as in the generalization, there is no condition on the positivity of certain coefficients in the recurrence relation: for the classical Favard version this is in contradiction with the existing literature. Furthermore, formulae for \(1/\Sigma_n |p_n^{(m)} (z) |^2\) \((m\) denoting the \(m\)-th derivative) are given in terms of the supremum over certain sets of real numbers connected a.o. with ratios of characteristic polynomials of certain classes of positive definite matrices of measures. Finally, an example using \((2N + 1)\)-banded matrices (the \(N\)-th power of a tridiagonal Jacobi matrix) is given.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15B57 Hermitian, skew-Hermitian, and related matrices
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