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Dyadic wavelets and refinable functions on a half-line
V. Yu. Protasova,
Yu. A. Farkovb a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Moscow State Geological Prospecting Academy
Abstract:
For an arbitrary positive integer
$n$ refinable functions on the positive
half-line
$\mathbb R_+$ are defined, with masks that are
Walsh polynomials of order
$2^n-1$. The Strang-Fix conditions, the partition of unity property,
the linear independence, the stability, and the orthonormality of integer
translates of a solution of the corresponding refinement equations are
studied. Necessary and sufficient conditions ensuring that these solutions generate
multiresolution analysis in
$L^2(\mathbb R_+)$ are deduced.
This characterizes all systems of dyadic compactly supported
wavelets on
$\mathbb R_+$ and gives one an algorithm for the
construction of such systems.
A method for finding estimates for the exponents of
regularity of refinable functions is presented,
which leads to sharp estimates in the case of small
$n$.
In particular, all the dyadic entire compactly supported refinable functions on
$\mathbb R_+$
are characterized. It is shown that a refinable function is either
dyadic entire or has a finite exponent of regularity, which, moreover, has
effective upper estimates.
Bibliography: 13 items.
UDC:
517.518.3+
517.965
MSC: Primary
42C40; Secondary
43A70 Received: 08.08.2005 and 26.07.2006
DOI:
10.4213/sm1126
Fulltext:
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