Symmetric Multivariate Orthogonal Refinable Functions

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 φ such that φ is symmetric about a point (centro-symmetric) and φ provides approximation order k, we show that φ 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 2Is-refinable function φ in any dimension such that φ is symmetric about a point and φ generates a classical coiflet. Finally, we prove that if a real-valued compactly supported orthogonal dyadic refinable function φ ∈ L2(Rs) has the axis symmetry, then φ cannot be a continuous function and φ 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, 46 (1998), 183–190] about symmetric orthogonal dyadic refinable functions. 2000 Mathematics Subject Classification. 42C20, 42C15, 41A25, 41A63.