# A characterization of boundary representations of positive matrices in the Hardy space via the Abel product

@article{Herr2019ACO, title={A characterization of boundary representations of positive matrices in the Hardy space via the Abel product}, author={John E. Herr and Palle E. T. Jorgensen and Eric S. Weber}, journal={Linear Algebra and its Applications}, year={2019} }

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 prove a characterization of those representing measures via a matrix identity by introducing a new operator product called the Abel Product.

## 5 Citations

White Noise Space Analysis and Multiplicative Change of Measures

- Physics, Mathematics
- 2021

In this paper we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space…

A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $$L^{2}$$ spaces

- Mathematics
- 2019

We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional,…

Reproducing kernels: Harmonic analysis and some of their applications

- Mathematics
- 2020

Abstract We develop a new harmonic analysis of reproducing kernels. Our approach is with view to a number of recent applications: We explore both geometric and algorithmic consequences of our…

Positive Definite Kernels, Algorithms, Frames, and Approximations

- Computer Science, MathematicsArXiv
- 2021

A new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space is proposed, designed with view to maximum likelihood solutions, minimization of “cost” problems, identification of principal components, and data-dimension reduction.

Harmonic analysis of network systems via kernels and their boundary realizations

- MathematicsDiscrete and Continuous Dynamical Systems - Series S
- 2021

<p style='text-indent:20px;'>With view to applications to harmonic and stochastic analysis of infinite network/graph models, we introduce new tools for realizations and transforms of positive…

## References

SHOWING 1-10 OF 38 REFERENCES

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

- Mathematics
- 2017

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…

Positive matrices in the Hardy space with prescribed boundary representations via the Kaczmarz algorithm

- MathematicsJournal 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…

Hilbert spaces of analytic functions, inverse scattering and operator models.II

- 1985

This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first, in conjunction…

On universality and convergence of the Fourier series of functions in the disc algebra

- MathematicsJournal d'Analyse Mathématique
- 2019

We construct functions in the disc algebra whose Fourier series are pointwise universal on countable and dense sets and their sets of divergence contain Gδ and dense sets and have Hausdorff dimension…

The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces

- Mathematics
- 2010

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each…

Analytic Reproducing Kernels and Factorization

- Mathematics
- 2004

This paper relates questions about factorizations of positive matrices to properties of analytic reproducing kernel Hilbert spaces. In particular the question of when the polynomials are dense in a…

Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem

- Mathematics
- 2013

We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison{Singer problem. The rst is Weaver’s conjecture…

Sub-Hardy Hilbert Spaces in the Unit Disk

- Mathematics
- 1994

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…