The Calderon Reproducing Formula And The Scaling Function Expansions Associated With The Heisenberg Group

This paper is concerned with Caldero′n reproducing formula and scaling functionexpansions in Lp(Hd) with 1 < p <∞. It consists of 3 chapters.In Chapter 1, the academic background, the purpose of our our research and themain work of the dissertation are introduced.In Chapter 2, the properties of Heisenberg group together with the Caldero′nreproducing formula converges to functions in L2(Hd) are studied. Furthermore, areproducing kernel is defined, then by the approximation of identity, a Caldero′n re-producing formula is established for functions in Lp(Hd) with 1 < p <∞.In Chapter 3, the multiscale analysis and the convergence properties of the char-acteristic function expansions in Lp(Hd) with 1 < p <∞are investigated. And ascaling function under the control of a radial function is given, then the convergenceproperties of such function expansions in Lp(Hd) (1 < p <∞) are studied.