Symmetric multivariate orthogonal refinable functions

Bin Han


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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