A~family of Nikishin systems with periodic recurrence coefficients

S. Delvaux ; A. López ; G. López Lagomasino

Sbornik. Mathematics, Tome 204 (2013) no. 1, pp. 43-74

Résumé

Suppose we have a Nikishin system of $p$ measures with the $k$th generating measure of the Nikishin system supported on an interval $\Delta_k\subset\mathbb R$ with $\Delta_k\cap\Delta_{k+1}=\varnothing$ for all $k$. It is well known that the corresponding staircase sequence of multiple orthogonal polynomials satisfies a $(p+2)$-term recurrence relation whose recurrence coefficients, under appropriate assumptions on the generating measures, have periodic limits of period $p$. (The limit values depend only on the positions of the intervals $\Delta_k$.) Taking these periodic limit values as the coefficients of a new $(p+2)$-term recurrence relation, we construct a canonical sequence of monic polynomials $\{P_{n}\}_{n=0}^\infty$, the so-called Chebyshev-Nikishin polynomials. We show that the polynomials $P_n$ themselves form a sequence of multiple orthogonal polynomials with respect to some Nikishin system of measures, with the $k$th generating measure being absolutely continuous on $\Delta_k$. In this way we generalize a result of the third author and Rocha [22] for the case $p=2$. The proof uses the connection with block Toeplitz matrices, and with a certain Riemann surface of genus zero. We also obtain strong asymptotics and an exact Widom-type formula for functions of the second kind of the Nikishin system for $\{P_{n}\}_{n=0}^\infty$. Bibliography: 27 titles.

Détail

Keywords: Nikishin system, block Toeplitz matrix, strong asymptotics, ratio asymptotics.
Mots-clés : multiple orthogonal polynomial, Hermite-Padé approximant