Meromorphic approximants to complex Cauchy transforms with polar singularities

Abstract

We study AAK-type meromorphic approximants to functions of the formwhere R is a rational function and λ is a complex measure with compact regular support included in (−1, 1), whose argument has bounded variation on the support. The approximation is understood in L p -norm of the unit circle, p ≥ 2. We dwell on the fact that the denominators of such approximants satisfy certain non-Hermitian orthogonal relations with varying weights. They resemble the orthogonality relations that arise in the study of multipoint Padé approximants. However, the varying part of the weight implicitly depends on the orthogonal polynomials themselves, which constitutes the main novelty and the main difficulty of the undertaken analysis. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of λ relative to the unit disk, that the approximants themselves converge in capacity to F , and that the poles of R attract at least as many poles of the approximants as their multiplicity and not much more.Mathematics Subject Classification (2000). primary 41A20, 41A30, 42C05; secondary 30D50, 30D55, 30E10, 31A15.

Asymptotic uniqueness of best rational approximants to complex Cauchy transforms in 𝐿² of the circle

“…The index theorem and the comparison criterion are expounded in section 4, paralleling the treatment for real Fourier coefficients given in [11]. Section 5 recalls the necessary material on interpolation from [15,13,31], which is needed to carry out the comparison between critical points and interpolants of lower degree required in the comparison criterion. Finally, elaborating on [11], we prove Theorem 1 in Section 6.…”

On Uniform Approximation of Rational Perturbations of Cauchy Integrals

“…(6.9) So, the asymptotic behavior of u n is governed by Theorem 7, applied with v n = q 2 n and h n = q n+m,m qw n+m , due to Lemma 8 and the fact that {w n } is a normal family in D none of which limit points has zeros. The latter was obtained in [9,Lem. 3.4] under the mere assumption that µ has infinitely many points in the support and an argument of bounded variation.…”

Source localization using rational approximation on plane sections

“…In this situation, the s k,p are both (triple) poles and branchpoints (of order 3/2) 3 . Yet, f p can be well approximated on the boundary T by a rational function with poles in D, see C and [Baratchart et al, 2006], [Baratchart and Yattselev, 2009]. …”

“…This result is used in [Baratchart and Yattselev, 2009] to study the behaviour of the poles of best rational approximants.…”

“…Further, deep convergence results from potential theory [Baratchart and Yattselev, 2009] assert that, for a function f p as in the present situation (which admits nitely many poles and branchpoints in D and has a smooth behaviour near T ), the m poles of R * m converge (in some weak sense) to the singularities (s k ) = (s k,p ) of f p as m increases (where, for notational simplicity, the index p is xed and has been omitted).…”