School Colloquium——The Pre-orthogonal Algorithm in Reproducing Kernel Hilbert Spaces
报告人：Prof. Tao Qian (Macau University of Science and Technology)
地点：Room 1114, Sciences Building No. 1
Abstract: A linear operator defined in the pattern of Riesz representation in a Hilbert space naturally introduces a reproducing kernel Hilbert space structure over the range space. The present study shows that such formulation of linear operators possesses a build-in mechanism of representing the solutions of most important types of fundamental problems, viz., the identification of the range, the inverse problem, and the Moore-Penrose pseudo-inverse problem. This talk aims to spell out the connections of these problems and gives explicit representation formulas in the form of infinite series of the solutions. Apart from the basic basis method, the talk mainly proposes a pre-orthogonal adaptive Fourier decomposition (POAFD) in contrast with the basis method. Optimality of the maximal selection principle of POAFD evidences that on the one-step-selection strategy the algorithm and its variations are indeed the most effective and offer practical and fast converging numerical solutions.