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Computing the Smoothness Exponent of a Symmetric Multivariate
Refinable Function

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Bin Han

## Abstract

Smoothness and symmetry are two important properties of a refinable
function. It is known that the Sobolev smoothness exponent of a refinable
function can be estimated by computing the spectral radius of certain
finite matrix which is generated from a mask. However, the increase of
dimension and the support of a mask tremendously increases the size of
the matrix and therefore make the computation very expensive. In this
paper, we shall present a simple algorithm to efficiently numerically
compute the smoothness exponent of a symmetric refinable function
with a general dilation matrix.
By taking into
account of symmetry of a refinable function,
our algorithm greatly reduces the size of the matrix
and enables us to numerically compute the Sobolev smoothness exponents
of a large
class of symmetric refinable functions.
Step by step numerically stable algorithms and details of the
numerical implementation are given.
To illustrate our results by performing some numerical experiments, we
construct a family of dyadic interpolatory masks in any dimension and
we compute the smoothness exponents of their refinable functions in
dimension three. Several examples will also be presented for computing
smoothness exponents of symmetric refinable functions on the quincunx
lattice and on the hexagonal lattice.

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