Frames and pseudo-inverses. (English) Zbl 0845.47002

Christensen, Ole

J. Math. Anal. Appl. 195, No. 2, 401-414 (1995).

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.

Reviewer: D.Przeworska-Rolewicz (Warszawa)

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

Keywords:

Bessel sequence; frame; pseudo-inverse operators; Moore-Penrose inverses; wavelet theory