MULTILEVEL DECOMPOSITION IN A HILBERT SPACE

Somdeb Majumdar (USA), Nhan Levan (USA) and Carlos S. Kubrusly (Brazil)

Received September 14, 2007

Abstract

By decomposing an element of a sequence – of Hilbert space bounded linear operators – into the sum of a lower level element and several higher level elements, one obtains a Multilevel Decomposition (MLD) of the element. Moreover, as we shall show, such a decomposition can result in a Multilevel Approximation (MLA) of vectors of the space. In particular, for the function space  the MLD of elements of a sequence of orthogonal projections  results in the well known Multiresolution Approximation (MRA) of Wavelet Theory. Also for the sequence of elements  where D is the -dyadic-scale operator, MLD also yields an approximation for functions of the space  An interesting feature of MLD is that it leads to a new interpretation of the Dilation-by-s operator as a “time-varying” shift operator.

Keywords and phrases: multilevel decomposition, multilevel approximation, multiresolution approximation, scaling operators, time-varying shifts on