# Mathematics Colloquium

## Professor Yiping Mao

speaks on

### Compact supported wavelet bases with group action invariance in higher dimension

Abstract. Wavelets are functions generated from one basic function by dilations and translations. They are used as analyzing tools, by both pure mathematicians (in harmonic analysis) and electrical engineers (in signal analysis). A particularly interesting development is the discovery of orthonormal (and biorthonormal) wavelets bases of $L^2(\Bbb{R}^n)$. The so-called subband coding schemes of perfect reconstruction (PR) is a general and efficient way to construct (bi)orthonormal bases of wavelets.
In this talk, we will discuss a class of subband coding schemes allowing
perfect reconstruction for signals sampled on the general lattices $\Gamma$ of $\Bbb{R}^n$ and describe the construction of compactly supported biorthonormal (or orthonormal) wavelet bases of $L^2(\Bbb{R}^n)$. The most interesting feature in our construction is that
the sets of generating functions $\{\psi_i\}_{i=0}^{k}$ for the synthesis and $\{\tilde{\psi}_i\}_{i=0}^{k}$ for the analysis, as well as the scaling
functions $\phi$ and $\tilde{\phi}$, are globally invariant under the action of a finite group generated by an operator $P: \Gamma^*\setminus \Bbb{R}^n \rightarrow \Gamma^*\setminus \Bbb{R}^n$. We classify all possible such group actions and present a algorithm to solve Bezout equation which is the key part of seeking PR subband coding schemes.

Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at Emmy's after the talk.
Host. Carolyn Gordon is the hostess. Anybody who is interested in having dinner with the speaker should contact Carolyn at 646-047.