On the convergence of orthorecursive expansions in nonorthogonal wavelets (Q1946426)

The paper deals with the so-called orthorecursive expansions (being a linear counterpart of `greedy' algorithms) considered in the general framework of non-orthogonal wavelets. The definition and basic properties of orthorecursive expansions are reported, and sufficient conditions under which a family of non-orthogonal wavelets is an unconditional orthorecursive expansion family in \(L^2(\mathbb R)\) (i.e. proper orthorecursive expansions converge to the expanded elements for any rearrangements inside a family) are provided. The idea of orthorecursive expansions is then generalized and discussed in the framework of linear operators in a normed space, and an unconditional recursive expansion family (producing convergent recursive series for any order of operators) in a linear normed space is specified. Proper sufficient convergence conditions of the resulting expansion series are given, in particular. Finally, stability with respect to the errors in the computing of the expansion coefficients for orthorecursive wavelet expansions is established. Complete proofs of the main results are included.