Abstract
Given a frame for a subspace W of a Hilbert space H, we consider all possible families of oblique dual frame vectors on an appropriately chosen subspace V. In place of the standard description, which involves computing the pseudoinverse of the frame operator, we develop an alternative characterization which in some cases can be computationally more efficient. We first treat the case of a general frame on an arbitrary Hilbert space, and then specialize the results to shift-invariant frames with multiple generators. In particular, we present explicit versions of our general conditions for the case of shift-invariant spaces with a single generator. The theory is also adapted to the standard frame setting in which the original and dual frames are defined on the same space.
Highlights
Frames are generalizations of bases which lead to redundant signal expansions [1,2,3,4]
The advantage of this characterization is that there is freedom in choosing the operator H so that it can be tailored such that the pseudoinverse of HT∗ is easier to compute than the pseudoinverse of TT∗
Given closed subspaces W and V of a Hilbert space H such that H = W ⊕ V⊥, the oblique projection EWV⊥ onto W along V⊥ is defined as the unique operator satisfying
Summary
Frames are generalizations of bases which lead to redundant signal expansions [1,2,3,4]. Our main result, derived, shows that the oblique dual frames can be characterized in an alternative way in which the pseudoinverse of TT∗ is replaced by the pseudoinverse of HT∗, where H is an appropriately chosen operator. The advantage of this characterization is that there is freedom in choosing the operator H so that it can be tailored such that the pseudoinverse of HT∗ is easier to compute than the pseudoinverse of TT∗. Before proceeding to the detailed development, we summarize the required mathematical preliminaries
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