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Symmetric multivariate orthogonal refinable functions

### Bin Han

## Abstract

In this paper, we shall investigate the symmetry property of a
multivariate orthogonal $M$-refinable function with a general
dilation matrix $M$. For an orthogonal $M$-refinable function
$\phi$ such that $\phi$ is symmetric about a point
(centro-symmetric) and $\phi$ provides approximation order $k$, we
show that $\phi$ must be an orthogonal $M$-refinable function that
generates a generalized coiflet of order $k$. Next, we show that
there does not exist a real-valued compactly supported orthogonal
$2 I_s$-refinable function $\phi$ in any dimension such that
$\phi$ is symmetric about a point and $\phi$ generates a classical
coiflet. Finally, we prove that if a real-valued compactly
supported orthogonal dyadic refinable function $\phi\in
L_2(\RR^s)$ has the axis symmetry, then $\phi$ cannot be a
continuous function and $\phi$ can provide approximation order at
most one. The results in this paper may provide a better picture
about symmetric multivariate orthogonal refinable functions. In
particular, one of the results in this paper settles a conjecture
in [D. Stanhill and Y. Y. Zeevi, IEEE Transactions on Signal
Processing, {\bf 46} (1998), 183--190] about symmetric orthogonal
dyadic refinable functions.

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