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.


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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