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: [email protected] (A. Borichev), [email protected] (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 -