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Frames and pseudo-inverses. (English) Zbl 0845.47002
A family $$\{f_i \}_{i\in I}$$ in an infinite-dimensional Hilbert space $${\mathcal H}$$ is called a Bessel sequence if $$\forall_{f\in {\mathcal H}} \sum_{i\in I} |\langle f,f_i \rangle|^2< \infty$$. A Bessel sequence $$\{f_i \}_{i\in I}$$ is called a frame if $\exists A>0:\;A|f|^2\leq \sum_{i\in I}|\langle f,f_i \rangle|^2, \qquad \forall f\in {\mathcal H}.$ There are studied connections between the frame theory and the theory of pseudo-inverse operators (i.e. Moore-Penrose inverses). According to the author, the frame theory is now a very useful tool in the wavelet theory.

MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 42C15 General harmonic expansions, frames
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