國立清華大學 數學系

Abstract: Inverse Problems deal with determining for a given input-output system an input that produces an observed output, or of determining an input that produces a desired output (or comes as close to it as possible), often in the presence of noise. Most inverse problems are ill-posed. Signal Analysis/Processing deals with digital representations of signals and their analog reconstructions from digital representations. Sampling expansions, filters, reproducing kernel spaces, various function spaces, and techniques of functional analysis, computational and harmonic analysis play pivotal roles in this area. Moment problems deal with recovery of a function or signal from its moments, and the construction of efficient stable algorithms for determining approximating the function. Again this is an ill-posed problem. Interrelated applications of inverse problems, signal analysis and moment problems arise, in particular, in image analysis and recovery and in many areas of science and technology. Several decades ago the connections among these areas (inverse problems, signal processing, and moment problems) was rather tenuous. Researchers in one of these areas were often unfamiliar with the techniques and relevance of the other two areas. The situation has changed drastically in the last 20 years. The common thread among inverse problems, signal analysis, and moment problems is a canonical problem: recovering an object (function, signal, picture) from partial or indirect information about the object. In this talk, we will provide perspectives on some aspects of this interaction with emphasis on ill-posed problems for operator and integral equations, and ill-posed problems in signal processing. We will show that function spaces, in particular reproducing kernel spaces and certain reproducing subspaces of L^p, play a pivotal role in this interaction. Tea Time: 3:30PM, R707