Weighted exponential approximation and non-classical orthogonal spectral measures

Alexander Borichev*, Mikhail Sodin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

A long-standing open problem in harmonic analysis is: given a non-negative measure μ on R{double-struck}, find the infimal width of frequencies needed to approximate any function in L2(μ). We consider this problem in the "perturbative regime", and characterize asymptotic smallness of perturbations of measures which do not change that infimal width. Then we apply this result to show that there are no local restrictions on the structure of orthogonal spectral measures of one-dimensional Schrödinger operators on a finite interval. This answers a question raised by V.A. Marchenko.

Original languageEnglish
Pages (from-to)2503-2545
Number of pages43
JournalAdvances in Mathematics
Volume226
Issue number3
DOIs
StatePublished - 15 Feb 2011

Funding

FundersFunder number
Agence Nationale de la Recherche

    Keywords

    • Orthogonal spectral measure
    • Schrödinger operator
    • Sturm-Liouville problem
    • Weighted exponential approximation

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