On higher order boundary derivatives of an analytic self-map of the unit disk

Abstract

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and sufficient conditions are given for the existence of a Schur-class function with the prescribed nontangential boundary expansion f (z) = s 0 + s 1 (z − t 0 ) + · · · + s N (z − t 0 ) N + o(|z − t 0 | N ) at a given point t 0 on the unit circle.

Schur Functions and Their Realizations in the Slice Hyperholomorphic Setting

“…The case of boundary interpolation is of special importance. See for instance [26] for related recent results.…”

Boundary Angular Derivatives of Generalized schur functions

“…In the case when n 0 = (N + 1)/2, this is enough to guarantee the existence of infinitely many rational functions s ∈ S satisfying conditions (2.1) (see [2] or [7]). The existence of such functions in the two remaining cases where n 0 = N/2 and γ n 0 ≥ 0 or where 0 < n 0 < N/2 and γ n 0 > 0 is guaranteed by [5,Theorem 2.3]. For every such s, the function f (z) = s(z)/z m solves the problem P N .…”

“…On the other hand, if f is such that for every b ∈ F B, the interpolation conditions (2.1) are satisfied by no Schur function, then the problem P N has no solutions. This simple idea allows us to reduce the problem P N to a similar problem for Schur-class functions the answer for which is known [5].…”

“…The Problems P N and P ∞ have been studied for Schur-class functions f ∈ S in [5] and [10], respectively. It was shown that the Problem P N has infinitely many solutions if and only if either | f 0 | < 1 or the maximal Hermitian matrix P f n 0 is positive definite and one of the cases (2)-(4) in Theorem 1.2 is in force.…”

“…The paper is organized as follows. In Section 2 we present the proof of Theorem 1.2 based on recent results [5] on Schur-class boundary interpolation. The proof of [5] Boundary angular derivatives 207 Theorem 1.3 is given in Section 3, at the end of which we also discuss the possible determinacy of the problem.…”