Mathematics Colloquia and Seminars

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Wavelets, framelets, and shapelets are three topics that emerged successively in applied and computational harmonic analysis. In this talk, I will describe our works in these three topics. A function is called a wavelet if its dyadic dilations and integral translations form an orthonormal basis. One often uses scaling functions to construct wavelets. Many authors researched the frequency domains of scaling functions. de Boor, DeVore, and Ron gave some necessary conditions that a domain is a frequency domain of some scaling function. Madych and Walter gave some sufficient conditions, respectively. I will talk about our result on the necessary and sufficient condition of frequency domains of scaling functions. I will also talk an interesting result that any ball cannot be a frequency domain of some scaling function. Framelets are a generalization of wavelets. Based on the unitary extension principle, Daubechies, Chui, and Ron constructed a lot of “nice” framelets. I will talk how to realize the peoriodization of framelets and construct periodic wavelet frames as well as the result on the optimal upper and lower bounds of periodic framelets. Shapelets are a new kind of wavelets. Recently, Refregier presented the concept of shapelets in astronomic image analysis. In view of the shape of a shaplet being a disk while that of a galaxy is an oriented ellipse, we suggest to modify shapelets such that their shapes are close to shapes of galaxies. Moreover we use modified shapelets to analyze the properties of galaxies. Modified shapelets would be expected to work very well in the future Large Synoptic Survey Telescope (LSST) imaging data.