TY - JOUR

T1 - Weighted exponential approximation and non-classical orthogonal spectral measures

AU - Borichev, Alexander

AU - Sodin, Mikhail

N1 - Funding Information:
The first named author was partially supported by the ANR projects DYNOP and FRAB. * Corresponding author. E-mail addresses: borichev@cmi.univ-mrs.fr (A. Borichev), sodin@post.tau.ac.il (M. Sodin).

PY - 2011/2/15

Y1 - 2011/2/15

N2 - 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.

AB - 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.

KW - Orthogonal spectral measure

KW - Schrödinger operator

KW - Sturm-Liouville problem

KW - Weighted exponential approximation

UR - http://www.scopus.com/inward/record.url?scp=78649450170&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.08.019

DO - 10.1016/j.aim.2010.08.019

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AN - SCOPUS:78649450170

SN - 0001-8708

VL - 226

SP - 2503

EP - 2545

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 3

ER -