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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
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