On the Tangential Interpolation Problem for H 2 Functions

Abstract

The aim of this paper is to solve a matrix-valued version of the Nevanlinna-Pick interpolation problem for Hi functions. We reduce this problem to a Nevanlinna-Pick interpolation problem for Schur functions and obtain a linear fractional transformation which describes the set of all solutions.

Representation Formulas for Hardy Space Functions Through the Cuntz Relations and New Interpolation Problems

“…The results in [5] are a special case of a family of interpolation problems with relaxed constraints. See [25,24].…”

“…and using Leech's factorization theorem (see next section), it was proved in [5] that f admits a (in general not unique) representation of the form (1.12)…”

“…For the readers convenience, we mention here briefly some of these connections. One motivation comes from earlier work [1] on two-sided, and tangential, interpolation for matrix functions; see also references [2] through [5], and [14]. We make additional connections also to to interpolation in de Branges-Rovnyak spaces [13], to wavelet filters, see e.g., [17], and to iterated function systems, see [18] by Courtney, Muhly, and Schmidt, and [28] by Rochberg, to Hardy classes [29,30] and to classical harmonic analysis; see e.g., [32,33,34,35].…”

“…In the present paper, we use a decomposition of elements in H 2 associated with isometries defined from b and which satisfy the Cuntz relation. The representation of Hardy functions, proved in [5], plays a major role in the reduction to interpolation problems in the setting of Schur functions. To ease the notation, we set the discussion in the framework of scalar-valued functions, but the paper itself (as well as [5]) is developed for matrix-valued functions.…”

“…The representation of Hardy functions, proved in [5], plays a major role in the reduction to interpolation problems in the setting of Schur functions. To ease the notation, we set the discussion in the framework of scalar-valued functions, but the paper itself (as well as [5]) is developed for matrix-valued functions. Besides being a key player in complex analysis, the Hardy space H 2 of the open unit disk plays an important role in signal processing and in the theory of linear dynamical systems.…”