Approximate oblique dual frames

TL;DR

This paper introduces the concept of approximate oblique dual frames in Hilbert spaces, providing theoretical foundations, characterizations, and practical examples, especially in shift-invariant spaces, to improve computational methods in frame analysis.

Contribution

It defines and investigates approximate oblique dual frames, offering new characterizations and conditions for their existence, with applications to shift-invariant spaces and B-spline generated frames.

Findings

Provides properties and characterizations of approximate oblique dual frames.

Establishes conditions for their existence in shift-invariant subspaces.

Demonstrates improved attributes of approximate duals over exact ones with B-spline examples.

Abstract

In representations using frames, oblique duality appears in situations where the analysis and the synthesis has to be done in different subspaces. In some cases, we cannot obtain an explicit expression for the oblique duals and in others there exists only one oblique dual frame which has not the properties we need. Also, in practice the computations are not exact. To give a solution to these problems, in this work we introduce and investigate the notion of approximate oblique dual frames first in the setting of separable Hilbert spaces. We present several properties and provide different characterizations of approximate oblique dual frames. We focus then on approximate oblique dual frames in shift-invariant subspaces of L^2(R)and g ive different conditions on the generators that assure their existence. The importance of approximate oblique dual frames from a numerical and computational point of view is illustrated with an example of frame sequences generated by B-splines, where the previous results are used to 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.