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   -> Volume 5, Issue 1

Preprints: Two papres on multi-wavelets from Strela
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Vasily Strela (

PostPosted: Tue Dec 03, 2002 4:15 pm    
Subject: Preprints: Two papres on multi-wavelets from Strela
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Preprints: Two papres on multi-wavelets from Strela

Title : Multiwavelets: Regularity, Orthogonality and Symmetry via
Two-scale Similarity Transform

Author : Vasily Strela,

Abstract: An important object in wavelet theory is the scaling
function $phi(t)$, satisfying a dilation equation $phi(t)=sum C_k
phi(2t-k)$. Properties of a scaling function are closely related to
the properties of the symbol or mask $P(omega)=sum C_k e^{-iomega
k}$. The approximation order provided by $phi(t)$ is the number of
zeros of $P(omega)$ at $omega = pi$, or in other words the number
of factors $(1+e^{-iomega})$ in $P(omega)$. In the case of
multiwavelets $P(omega)$ becomes a matrix trigonometric
polynomial. The factors $(1+e^{-iomega})$ are replaced by a matrix
factorization of nn $P(omega)$, which defines the approximation order
of the multiscaling function. This matrix factorization is based on
the two-scale similarity transform (TST). In this paper we study
properties of the TST and show how it is connected with the theory of
multiwavelets. This approach leads us to new results on regularity,
symmetry and orthogonality of multi-scaling functions and opens an
easy way to their construction.

Title : Construction of Multi-Scaling Functions with
Approximation and Symmetry

Authors : Gerlind Plonka,
Vasily Strela,

Abstract: This paper presents a new and efficient way to create
multi-scaling functions with given approximation order, regularity,
symmetry and short support. Previous techniques were operating in the
time domain and required the solution of large systems of nonlinear
equations. By switching to the frequency domain and employing the
latest results of the multiwavelet theory we were able to elaborate a
simple and efficient method of construction of multi-scaling
functions. Our algorithm is based on a recently found factorization
of the refinement mask through the two-scale similarity transform
(TST). Theoretical results and new examples

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