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A long-standing open problem in harmonic analysis is: given a non-negativemeasure $\mu$ on $\mathbb R$, find the infimal width of frequencies needed toapproximate any function in $L^2(\mu)$. We consider this problem in the"perturbative regime", and characterize asymptotic smallness of perturbationsof measures which do not change that infimal width. Then we apply this resultto show that there are no local restrictions on the structure of orthogonalspectral measures of one-dimensional Schrodinger operators on a finiteinterval. This answers a question raised by V.A.Marchenko.

Weighted exponential approximation and non-classical orthogonal spectral measures