On the determinacy problem for measures

Mitkovski, Mishko; Poltoratski, Alexei

doi:10.1007/s00222-015-0588-6

Cited by 7 publications

(12 citation statements)

References 16 publications

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“…From one of the results of [33] (or Theorem 17, Chapter 4, in [41]) we deduce Corollary 2. If W is a weight, A is a separated sequence and B is any closed set then sup{a| M W a (A, B) = ∅} ≤ 2πD * (A).…”

Section: The Following Version Of the Type Theorem Is Theorem 36 Frommentioning

confidence: 80%

“…The following statement is a combination of Lemmas 13 and 16 from [33]. (A, B) is non-empty it contains a discrete measure ν, whose positive and negative parts have interlacing supports.…”

Section: Completeness Gap and Type Problemsmentioning

confidence: 97%

“…The Toeplitz approach, pioneered by Hruschev, Nikolski and Pavlov in [36,19], was further developed and applied to a broader class of problems in [24,25]. Recent applications of the Toeplitz approach in [32,39,40,37,33,41] involved a variety of problems of UP and adjacent fields. Two of such problems, the so-called Gap and Type Problems, will be revisited in this note.…”

Section: Introductionmentioning

confidence: 99%

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Two-spectra theorem with uncertainty

Makarov

Poltoratski

2019

J. Spectr. Theory

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The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type Problems of UP we prove a version of Borg's two-spectra theorem for Schrödinger operators, allowing uncertainty in the placement of the eigenvalues. We give a formula for the exact 'size of uncertainty', calculated from the lengths of the intervals where the eigenvalues may occur. Among other applications, we describe pairs of indeterminate operators in the three-interval case of the mixed spectral problem. At the end of the paper we discuss further questions and open problems.

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Section: The Following Version Of the Type Theorem Is Theorem 36 Frommentioning

confidence: 80%

“…The following statement is a combination of Lemmas 13 and 16 from [33]. (A, B) is non-empty it contains a discrete measure ν, whose positive and negative parts have interlacing supports.…”

Section: Completeness Gap and Type Problemsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Two-spectra theorem with uncertainty

Makarov

Poltoratski

2019

J. Spectr. Theory

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“…Very recently, the second author jointly with Poltoratski, refined these results even further [37], and obtained a generalization of the Beurling spectral gap theorem that strengthened this theorem by a factor of two.…”

Section: Spectral Gap and Oscillationmentioning

confidence: 96%

Kernels of Toeplitz operators

Hartmann¹,

Mitkovski²

2016

Recent Progress on Operator Theory And Approximation in Spaces of Analytic Functions

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Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we will focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and Pólya sequences. Second, if the kernel is nontrivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.2010 Mathematics Subject Classification. 30J05, 30H10, 46E22.

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“…The class of MIFs serves as a natural object in the so called Toeplitz approach to the Uncertainty Principle, see ([19], [20], [27]). It was used to obtain an extension of the Beurling-Malliavin theory ( [19], [20]) and to study classical problems of Fourier analysis ( [22], [23], [25], [26]). Some function theoretic questions arising from such applications were treated in [30].…”

Section: Introductionmentioning

confidence: 99%