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Symmetry property and construction of wavelets
with a general dilation matrix

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Bin Han

## Abstract

In this note, we are interested in the symmetry property of a
refinable function with a general dilation matrix. We investigate the
symmetry group of a mask so that its associated refinable function
with a general dilation matrix has a certain kind of
symmetry. Given two dilation matrices which produce the same lattice,
we demonstrate that if a mask has a certain kind of symmetry, then its
associated refinable functions with respect to the two dilation
matrices are the same; therefore, the two corresponding derived
wavelet systems are essentially the same.
Finally, we illustrate that for any dilation matrix, orthogonal masks,
as well as interpolatory masks having nonnegative symbols,
can be easily constructed with any preassigned order of sum rules
by employing a linear transform. Without solving any equations, the
method in this note on constructing masks with certain desirable
properties is simple, painless and general. Examples of quincunx
wavelets and wavelets with respect to the checkerboard lattice are
presented to illustrate the general theory.

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