Interpolation by derivatives in \(H^{\infty}\)
type: Item
instance of: scholarly article
title: Interpolation by derivatives in \(H^{\infty}\)
zbMATH Open document ID: 1399.30144
published in: Acta Mathematica Hungarica
publication date: 2018-02-12T00:00:00Z
review text: One of the main problems studied in the paper concerns the simultaneous interpolation by the derivatives \(f^{(n)}\) and \(f^{(m)}\), of a bounded, analytic function \(f\) on the unit disk. Given a sequence \(C=(c_k)\) of points of the disk, the authors introduce a weighted \(\ell^\infty\) class of complex sequences \[ \Lambda_k(C):=\{(\lambda_j)_{j=1}^\infty: \quad \sup_j | \lambda_j| (1-| c_j| ^2)^k<\infty \}. \] Given two disjoint sequences \(A=(a_j)\), \(B=(b_j)\) in the unit disk, their union \(Z=A\cup B\) is said to be \((n,m)\)-interpolating with respect to \((A,B)\) if for each \((\alpha_j)\in\Lambda_n(A)\) and \((\beta_j)\in\Lambda_m(B)\) there is a function \(f\in H^\infty\) so that \[ f^{(n)}(a_j)=\alpha_j, \qquad f^{(m)}(b_j)=\beta_j, \qquad j=1,2,\ldots. \] The main result of the paper claims that \(Z=A\cup B\) is \((n,m)\)-interpolating with respect to \((A,B)\) if and only if both \(A\) and \(B\) are uniformly separated.
Mathematics Subject Classification ID: 30E05
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30J10
zbMATH DE Number: 6837282
zbMATH Keywords: interpolating sequence
zbMATH Keywords: bounded analytic function
zbMATH Keywords: uniformly separated sequence
zbMATH Keywords: Carleson measure
author: Francesc Tugores
author: Laia Tugores
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
full work available at URL: s10474-017-0772-9
OpenAlex ID: W2766600677
cites work: Interpolating sequences for the derivatives of Bloch functions
cites work: Interpolation by functions in the Bloch space
cites work: Free interpolation for holomorphic functions regular to the boundary
cites work: An Interpolation Problem for Bounded Analytic Functions
cites work: Interpolating sequences for functions satisfying a Lipschitz condition
cites work: Characterisations for analytic functions of bounded mean oscillation
cites work: Beurling type density theorems in the unit disk
DOI: 10.1007/S10474-017-0772-9
Recommended article: Q3709321
Recommended article: Interpolation by Differences In H∞
Recommended article: Q3692302
Recommended article: Interpolation for a subclass of \(H^\infty\)
Recommended article: Q5322221
Recommended article: Generalized Interpolation in H ∞
Recommended article: Interpolation in A −∞
Recommended article: H∞ Interpolation Into A Class Of Analytic Functions
Recommended article: Free interpolation by functions with derivatives in \(H^ 1\)
Recommended article: Interpolation in ∊∘′
description: scientific article; zbMATH DE number 6837282