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   -> Volume 5, Issue 7

Preprint: Accuracy of multidimensional refinable functions.
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Chris Heil (

PostPosted: Fri Aug 23, 1996 2:37 pm    
Subject: Preprint: Accuracy of multidimensional refinable functions.
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#2 Preprint: Accuracy of multidimensional refinable functions.

The following preprint is available.

C. Cabrelli, C. Heil, and U. Molter:

Accuracy of lattice translates of several multidimensional refinable

ABSTRACT. Complex-valued functions $f_1,dots,f_r$ on ${old R}^d$
are {it refinable} if they are linear combinations of finitely many
of the rescaled and translated functions $f_i(Ax-k)$, where the
translates $k$ are taken along a lattice $Gamma subset {old R}^d$
and $A$ is a {it dilation matrix} that expansively maps $Gamma$
into itself. Refinable functions satisfy a {it refinement equation}
$f(x) = sum_{k in Lambda} c_k , f(Ax-k)$, where $Lambda$ is a
finite subset of $Gamma$, the $c_k$ are $r imes r$ matrices, and
$f(x) = (f_1(x),dots,f_r(x))^{ ext{T}}$. The {it accuracy} of $f$
is the highest degree $p$ such that all multivariate polynomials $q$
with degree$(q) < p$ are exactly reproduced from linear combinations
of translates of $f_1,dots,f_r$ along the lattice $Gamma$. In this
paper, we determine the accuracy $p$ from the matrices $c_k$.
Moreover, we determine explicitly the coefficients $y_{alpha,i}(k)$
such that $x^alpha = sum_{i=1}^r sum_{k in Gamma}
y_{alpha,i}(k) , f_i(x+k)$. These coefficients are multivariate
polynomials $y_{alpha,i}(x)$ of degree $|alpha|$ evaluated at
lattice points $k in Gamma$.

Postscript for this preprint is available by anonymous ftp
from the address

(or for compressed postscript). I will email postscript
or send hardcopy upon request.

Other papers on the topics of wavelets and time-frequency analysis
are also available in the same directory. There is a README file
that lists the titles available.

Chris Heil (
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