Perturbation of frame sequences and approximate reconstructions

SIGRID HEINEKEN; EWA MATUSIAK; VICTORIA PATERNOSTRO

Córdoba

Congreso; CLAM 2012, IV Congreso Latinoamericano de Matemáticos; 2012

We study perturbation of frame sequences in $L^2(mathbb{R})$. In particular, we consider frames sequences of irregular translates. Perturbation theory of frame sequences differs from Riesz sequences by additional difficulty of computing the gap between the spaces the respective sequences span. On one side, we analyze a perturbation of the generator and on the other side a perturbation of the set of points along which we translate. For both, we estimate the gap between the space spanned by the original frame sequence and the perturbed one in order to obtain conditions for the new sequence to be a frame sequence. We also study the canonical duals of the perturbed frame sequences and show that, under certain conditions, if the perturbation of the frame sequence is small the same happens for the canonical duals. With this we estimate the error in the approximation of functions belonging to the perturbed space.