Approximate oblique dual frames

DÍAZ J. P.; HEINEKEN S. B.; MORILLAS P. M.

APPLIED MATHEMATICS AND COMPUTATION

ELSEVIER SCIENCE INC

Lugar: Amsterdam; Año: 2023 vol. 452

0096-3003

In representations using frames, oblique duality appears in situations where theanalysis and the synthesis has to be done in different subspaces. In some cases, we cannotobtain an explicit expression for the oblique duals and in others there exists only one obliquedual frame which has not the properties we need. Also, in practice the computations are notexact. To give a solution to these problems, in this work we introduce and investigate thenotion of approximate oblique dual frames first in the setting of separable Hilbert spaces. Wepresent several properties and provide different characterizations of approximate oblique dualframes. We focus then on approximate oblique dual frames in shift-invariant subspaces of L2 (R)and give different conditions on the generators that assure their existence. The importance ofapproximate oblique dual frames from a numerical and computational point of view is illustratedwith an example of frame sequences generated by B-splines, where the previous results are usedto construct approximate oblique dual frames which have better attributes than the exact ones.We provide an expression for the approximation error and study its behaviour.