Compactly Supported Orthonormal Complex M-Wavelets with Symmetry

(Abstract)

In this talk, we shall present some recent results and constructions on compactly supported orthonormal complex wavelets with symmetry. It is well-known that except the discontinuous Haar wavelet, compactly supported real-valued orthonormal wavelets cannot have symmetry. In this talk, we first study symmetric orthonormal dyadic complex wavelets such that the orthonormal refinable functions have high linear-phase moments and the wavelets have high vanishing moments. Such wavelets lead to real-valued symmetric tight wavelet frames with desirable moment properties, and are related to real-valued coiflets which are of interest in numerical algorithms. Then we shall address symmetric orthonormal complex wavelets with a general dilation factor M (that is, M-wavelets). This problem is related to the interesting matrix extension problem with symmetry, which plays a fundamental role in many areas such engineering, wavelets, and mathematics.

We shall present two families of compactly supported symmetric orthonormal complex $M$-wavelets with arbitrarily high vanishing moments or arbitrarily high linear-phase moments. Connections of complex wavelets to multiwavelets will be mentioned.