## Pseudoframes for subspaces with applications.(English)Zbl 1058.42024

Given a subspace $$X$$ of a Hilbert space $$H$$, a Bessel sequence $$\{x_n\}$$ is said to be a pseudoframe for $$X$$ w.r.t. a Bessel sequence $$\{x_n^*\}$$ if $$f= \sum \langle f, x_n^*\rangle x_n$$ holds for all $$f\in X$$. Pseudoframe decompositions are more general than classical frame decompositions: $$\{x_n\}$$ do not necessarily belong to $$X$$ and might not be a frame. In the paper, pseudoframes are characterized in terms of operators, and the issue of finding duals is discussed in detail. Pseudoframes are considered in shift-invariant spaces, and applications to signal restoration and noise reduction are sketched.

### MSC:

 42C15 General harmonic expansions, frames 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

### Keywords:

pseudoframes; frames; series expansion
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