Weighted exponential approximation and non-classical orthogonal spectral measures

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

2010-04-11

arxiv.org/abs/1004.1795v3

Advances in Mathematics 226 (2011) 2503-2545

Mathematics

Classical Analysis and ODEs

footnote 4 is corrected; some changes are made in the proof of Theorem 2.10

Scientific paper

A long-standing open problem in harmonic analysis is: given a non-negative measure $\mu$ on $\mathbb R$, find the infimal width of frequencies needed to approximate any function in $L^2(\mu)$. 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 Schrodinger operators on a finite interval. This answers a question raised by V.A.Marchenko.

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