Approximate oblique duality for frames in shift-invariant spaces

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

Buenos Aires

Workshop; Workshop in Harmonic Analysis, Sampling Theory, Machine Learning, and Data Science; 2022

In representations using frames, oblique duality appears insituations where the analysis and the synthesis has to be done indifferent subspaces. In some cases, we cannot obtain an explicitexpression for the oblique duals and in others there exists only oneoblique dual frame which has not the properties we need. Also, inpractice the computations are not exact. To give a solution to theseproblems, we introduce and investigate the notion ofapproximate oblique dual frames first in the setting of separableHilbert spaces. We focus then on approximate oblique dual frames in shift-invariant subspaces of$Lebcuad$ and give different conditions on the generators thatassure their existence. The importance of approximate oblique dualframes from a numerical and computational point of view isillustrated with an example of frame sequences generated by$B$-splines, where the previous results are used to constructapproximate oblique dual frames which have better attributes thanthe exact ones. We provide an expression for the approximation errorand study its behaviour.