Mixed type Hermite-Padé approximation inspired by the Degasperis-Procesi equation

Abstract

In this work we present new results on the convergence of diagonal sequences of certain mixed type Hermite-Padé approximants of a Nikishin system. The study is motivated by a mixed Hermite-Padé approximation scheme used in the construction of solutions of a Degasperis-Procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string.

Asymptotic of Cauchy Biorthogonal Polynomials

“…The definition of a Nikishin system can be extended to the case when the intervals ∆ j are unbounded or touching. The results of this section remain valid when the Nikishin systems are constructed following the more general definition given in [13]. However, the asymptotic results given in Sections 3-4 require that we use the more restricted version presented here (which, incidentally, coincides with its original formulation in [15]).…”

“…, m, is obtained solving a homogeneous linear system of (n + 1)m equations on the (n + 1)m + 1 coefficients of the polynomials. Among other properties, in [13] (see also Lemma 2.4 below) it was shown that deg a n,m = n and the vector of polynomials (a n,0 , . .…”

“…Some of the statements of the next two results may be extracted from [13]. However, new notation is introduced and several formulas do not appear explicitly in that paper so, for convenience of the reader, we include a full proof.…”

On the Convergence of Multi-Level Hermite–Padé Approximants for a Class of Meromorphic Functions

“…Such areas cover, but are not limited to, number theory, non-intersecting brownian motions, multiple orthogonal polynomial ensembles, random matrix theory, and differential equations. For specific references see [12]. In particular, in [13] they were used to find discrete (peakon) solutions of the Degasperis-Procesi partial differential equation (see also references therein).…”

“…We wish to point out that the pair of measures appearing in the problem form a generator of a Nikishin system (see definition in next subsection). For more details see [13] and [12,Appendix].…”