On Uniform Approximation of Rational Perturbations of Cauchy Integrals

Abstract

Let [c, d] be an interval on the real line and µ be a measure of the form dµ =μdω [c,d] with an argument of bounded variation, and ω [c,d] is the normalized arcsine distribution on [c, d]. Further, let p and q be two polynomials such that deg(p) < deg(q) and [c, d] ∩ z(q) = ∅, where z(q) is the set of the zeros of q. We show that AAK-type meromorphic as well as diagonal multipoint Padé approximants toconverge locally uniformly to f in D f ∩ D and D f , respectively, where D f is the domain of analyticity of f and D is the unit disk. In the case of Padé approximants we need to assume that the interpolation scheme is "nearly" conjugate-symmetric. A noteworthy feature of this case is that we also allow the densityμ to vanish on (c, d), even though in a strictly controlled manner.Mathematics Subject Classification (2000). 42C05, 41A20, 41A21, 41A30.

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

“…In order to use this criterion, we need an appraisal of the error in interpolation to f by members of R n and R n−1 . In the next section, we gather the necessary estimates for the class of functions introduced in Theorem 1 after the works [13,31].…”

“…Next, a criterion for being a local minimum is set up, based on a comparison between the error function generated by the critical point under examination and the error function attached to a particular multipoint Padé interpolant of lower degree; we call it for short the comparison criterion. Finally, the fact this criterion applies when the degree is sufficiently large depends on strong asymptotic formulas for the error in rational interpolation to functions of the type we consider, that were recently obtained in [15,13,31], and on a specific design of interpolation nodes to build the particular multipoint Padé interpolant of lower degree that we need.…”

“…Developing the precise estimates would make the paper heavier (compare [13,Thm. 4] and [31,Thm. 5]), so we felt better omitting this stronger version.…”

“…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.…”

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

“…The appearance of the normal family {θ n } in (2.16) is not necessitated by Theorem 2 but is included for possible application to meromorphic approximation [53].…”