# Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle

@inproceedings{Makarov2005MeromorphicIF, title={Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle}, author={Nikolai G. Makarov and Alexei Poltoratski}, year={2005} }

This paper touches upon several traditional topics of 1D linear complex analysis
including distribution of zeros of entire functions, completeness problem for
complex exponentials and for other families of special functions, some problems
of spectral theory of selfadjoint differential operators. Their common
feature is the close relation to the theory of complex Fourier transform of
compactly supported measures or, more generally, Fourier–Weyl–Titchmarsh
transforms associated with…

## 63 Citations

Asymptotically orthonormal basis and Toeplitz operators

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

Uniform boundedness of the derivatives of meromorphic inner functions on the real line

- Mathematics
- 2013

Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions…

Basis properties of complex exponentials and invertibility of Toeplitz operators

- Mathematics
- 2018

We give a criterion for basicity of a sequence of complex exponentials in terms of the invertibility properties of a certain naturally associated Toeplitz operator. The criterion is similar to the…

Restricted interpolation by meromorphic inner functions

- Mathematics
- 2016

Abstract Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent…

Spectral theory of rank one perturbations of normal compact operators

- MathematicsSt. Petersburg Mathematical Journal
- 2019

We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study…

Vector Semi-Fredholm Toeplitz Operators and Mean Winding Numbers

- MathematicsNagoya Mathematical Journal
- 2009

Abstract For a continuous nonvanishing complex-valued function g on the real line, several notions of a mean winding number are introduced. We give necessary conditions for a Toeplitz operator with…

Etudes for the inverse spectral problem

- Mathematics
- 2022

In this note we study inverse spectral problems for canonical Hamiltonian systems, which encompass a broad class of second order differential equations on a half-line. Our goal is to extend the…

Uniqueness theorems for meromorphic inner functions

- Mathematics
- 2021

We prove some uniqueness problems for meromorphic inner functions on the upper half-plane. In these problems we consider spectral data depending partially or fully on the spectrum, derivative values…

## References

SHOWING 1-10 OF 44 REFERENCES

Inner functions and related spaces of pseudocontinuable functions

- Mathematics
- 1993

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…

Inverse spectral problems and closed exponential systems

- Mathematics
- 2005

Consider the inverse eigenvalue problem of the Schr?odinger operator de- fined on a finite interval. We give optimal and almost optimal conditions for a set of eigenvalues to determine the…

Analysis of Toeplitz Operators

- Mathematics
- 1991

A revised introduction to the advanced analysis of block Toeplitz operators including recent research. This book builds on the success of the first edition which has been used as a standard reference…

m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

- Mathematics
- 1997

We study inverse spectral analysis for finite and semi-infinite Jacobi matricesH. Our results include a new proof of the central result of the inverse theory (that the spectral measure determinesH).…

Operators, Functions, and Systems: An Easy Reading

- Mathematics
- 2002

Together with the companion volume by the same author, Operators, Functions, and Systems: An Easy Reading. Volume 2: Model Operators and Systems, Mathematical Surveys and Monographs, Vol. 93, AMS,…

Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum

- Mathematics
- 1999

We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential q of a one-dimensional Schrodinger operator H =…

On the closure of characters and the zeros of entire functions

- Mathematics
- 1967

The problem to be studied in this paper concerns the closure properties on an interval of a set of characters {e~nx}~, where A = {2n}~ is a given set of real or complex numbers without finite point…

Zero sets and multiplier theorems for star-invariant sub spaces

- Mathematics
- 2002

Given an inner function θ, let {Kskθ/p}:= Hp ∩θ {Hsk0/p} be the corresponding star-invariant subspace of the Hardy spaceHp. We show that, unless θ is a finite Blaschke product, the zero sets for…

The Logarithmic Integral

- Mathematics
- 2001

In this chapter we discuss the argument principle and develop several of its consequences. In Section 1 we derive the argument principle from the residue theorem, and we use the argument principle to…

On bounded analytic functions

- Mathematics
- 1950

The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2].1 While it is admitted that the main idea…