A higher order analogue of the Carathéodory–Julia theorem

Abstract

A higher order analogue of the classical Carathéodory-Julia theorem on boundary derivatives is proved.

Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel

“…Using hypergeometric series we show that this sum is equal to ±2 n , where the choice of sign depends on r. We mention a recent and very interesting work of V. Bolotnikov and A. Kheifets [6] who obtained an analogue of the classical Carathéodory-Julia theorem on boundary derivatives. Using different techniques, the authors also obtained a condition which guarantees that we can write an analogue of formula (1.2) for the de Branges-Rovnyak spaces H(b).…”

Carathéodory–Julia type conditions and symmetries of boundary asymptotics for analytic functions on the unit disk

“…We start with the classical Carathéodory-Julia theorem [7,9]. The following are equivalent: Higher order analogues of the above results have been presented in [3]. To recall them we first introduce some needed notations and definitions.…”

“…Note that the structured matrix P w n (t 0 ) (which first appeared in [11]) is a higher order analogue of the product w 1 t 0 w * 0 . It turns out (and will be shown in this paper) that property (3) in Theorem 1.4 follows from a weaker assumption |w 0 (t 0 )| = 1 and P w n (t 0 ) = P w n (t 0 ) * , (1.17)…”

“…The agreeing of the two asymptotics was actually taken by I.V. Kovalishina as a basis for her constructions in [11], [11] as opposed to [3] and [6], where all the constructions started from (1.15).…”

“…Since |f 0 | = 1 and P f k−1 > 0, it follows by an interpolation result (see e.g., [11,1,3]), there exists a finite Blaschke product b(z) such that…”