Strong asymptotics of polynomials orthogonal with respect to a complex weight

Padé approximants, continued fractions, and orthogonal polynomials

“…where ρ is a complex function holomorphic on [−1, 1] and ρ(x) ̸ = 0, the behaviour of the m poles of [n/n] f in a small neighbourhood of a pole of r of multiplicity m 3 can be described more precisely: asymptotically, they lie at the vertices of a regular m-gon (see [96], and also [97]). These results hold for the simplest algebraic functions of the form…”

“…Several generalizations of the classical Padé approximants have been considered: multipoint Padé approximants, Padé approximants to orthogonal expansions, to Faber series and trigonometric series, Hermite-Padé approximation, and others (see [8], [11], [12], [26], [30], [31], [97], [123], [124], [132]- [139]). All these cases are concerned with approximation by rational functions with free poles, and the approximants themselves have a non-linear character.…”

“…Convergence of Padé approximants for hyperelliptic functions 2.1. In the mid-1970s Nuttall obtained another complex analogue of the Markov theorem in a series of papers [33], [35], [36] (see also [40], [97]- [99]), this time for general hyperelliptic functions. Instead of several intervals on the real line he considered the situation when the support of the measure is a union of several pairwise disjoint arcs in the complex plane.…”

Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval

“…The weight (1.1) is a particular case of that analysis, although its simplicity allows for more explicit results. It is also important to mention that polynomials with respect to a complex exponential weight have also been considered recently in the work of Suetin [28]. In this reference the author studies (using different techniques) the case ω = 1 in our notation, modified with a Chebyshev factor (1 − x 2 ) −1/2 .…”

Аппроксимации Паде, Непрерывные Дроби И Ортогональные Многочлены

“…3, полюса r уточняется следующим образом: асимптотически они располагаются в вершинах правильного m-угольника (см. [96], а также [97]). Приведенные результаты справедливы для простейших алгебраических функций вида…”