Perturbation of orthonormal bases inL 2-spaces

The paper improves and generalizes a classical result from Paley and Wiener in their book on Fourier transforms. Paley and Wiener gave conditions on functionsh _n that imply that the sequence (1+h _n (x))e ^inx is a Riesz basis forL ²–,]. These conditions involve theL ²-norm of the second derivativesh _n . The new results replace the differential operatoryy by more general differential operators inL ²-spaces, in particular, by the Hermite differential operator inL ²(R), ande ^inx by arbitrary orthonormal bases.