# A matrix characterization of boundary representations of positive matrices in the Hardy space

@article{Herr2017AMC,
title={A matrix characterization of boundary representations of positive matrices in the Hardy space},
author={John E. Herr and Palle E. T. Jorgensen and Eric S. Weber},
journal={arXiv: Functional Analysis},
year={2017}
}
• Published 9 May 2017
• Mathematics
• arXiv: Functional Analysis
Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We introduce a potential characterization of those measures via a matrix identity and show that the characterization holds in several important…
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## References

SHOWING 1-10 OF 19 REFERENCES
Positive matrices in the Hardy space with prescribed boundary representations via the Kaczmarz algorithm
• Mathematics
Journal d'Analyse Mathématique
• 2019
For a singular probability measure $\mu$ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation on the unit circle with respect to $\mu$.
Hilbert spaces of analytic functions, inverse scattering and operator models.I
• Mathematics
• 1984
This paper develops a method for obtaining linear fractional representations of a givenn×n matrix valued function which is analytic and contractive in either the unit disc or the open upper half
Basic Boundary Interpolation for Generalized Schur Functions and Factorization of Rational J-unitary Matrix Functions
• Mathematics
• 2006
We define and solve a boundary interpolation problem for generalized Schur functions s(z) on the open unit disk $$\mathbb{D}$$ which have preassigned asymptotics when z from $$\mathbb{D}$$ tends
Sub-Hardy Hilbert Spaces in the Unit Disk
Hilbert Spaces Inside Hilbert Spaces. Hilbert Spaces Inside H 2 . Cauchy Integral Representations. Nonextreme Points. Extreme Points. Angular Derivatives. Higher Derivatives. Equality of H(b) and
Dense analytic subspaces in fractalL2-spaces
• Mathematics
• 1997
We show that for certain self-similar measures μ with support in the interval 0≤x≤1, the analytic functions {ei2πnx:n=0,1,2, …} contain an orthonormal basis inL2 (μ). Moreover, we identify subsetsP ⊂
ON THE BEURLING DIMENSION OF EXPONENTIAL FRAMES
• Mathematics
• 2010
Abstract We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling
Inner functions and related spaces of pseudocontinuable functions
Let θ be an inner function, let α ∈ C, ¦α¦=1. Then the harmonic function ℜ[(α+θ)]/(α−θ)] is the Poisson integral of a singular measureσα D. N. Clark's known theorem enables us to identify in a
Affine fractals as boundaries and their harmonic analysis
• Mathematics
• 2010
We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we
THE ART OF FRAME THEORY
The theory of frames for a Hilbert space plays a fundamental role in signal processing, image processing, data compression, sampling theory and more, as well as being a fruitful area of research in
Theory of Reproducing Kernels.
Abstract : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie