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      1. Isomorphism in wavelets


        【講座題目】Isomorphism in wavelets



        【主 講 人】戴興德 教授


        戴興德,美國德克薩斯A&M大學獲得博士學位,現為美國北卡羅來納夏洛特分校終身教授。戴教授是國際著名的泛函分析與小波分析專家,他和導師D. Larson教授于上世紀90年代成功地將泛函分析算子代數與調和分析框架小波理論相結合,開創了這兩個領域交叉研究的新方向,該理論具有著廣泛的理論和應用價值,是國際研究的熱點領域,被國際上稱為Dai-Larson理論。戴教授的開創性的研究成果(高倍引論文)為 Wandering vectors for unitary systems and orthogonal wavelets (Mem. Amer. Math. Soc.134 (1998), no. 640, viii+68 pp);  Wavelet sets in Rn ( J. Fourier Anal. Appl.3 (1997), no. 4, 451–456);  Wavelet sets in rn II (Amer. Math. Soc., Providence, RI, 1998) 。


        Two scaling functions and for Parseval frame wavelets are algebraically isomorphic,  , if they have matching solutions to their (reduced) isomorphic systems of equations.

        Let A and B be d×d and s×s dyadic expansive integral matrices with d, s≥1  respectively and let   be a scaling function associated with matrix A and generated by a finite solution. Then there always exists a scaling function   associated with matrix B such that    is algebraically isomorphic to  .

        An example shows that the assumption on the finiteness of the solutions can not be removed.