Current Development in Theory and Applications of Wavelets
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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
