Generalized sampling expansion in Riesz basis and frame setting

Abstract

This dissertation handles the sampling expansion in Riesz basis and frame setting. Sampling theory enables us to convert an analog signal (continuous function) into a digital signal (discrete function) without loss of information of signal so that it plays an important role in communication engineering, signal processing and so on. Hence many mathematicians and engineers have produced many interesting results in sampling theory. First of all, in 1949, Shannon presented the sampling theorem for finite energy band-limited signals which gave a good foundation of sampling theory and made a big influence in application of sampling theory. Later on, with the development of multi-channel system and multi-input multi-output(MIMO) system, many researchers presented the sampling theory about those. But, many existing result on channeled sampling has some problems in stability or well-posedness. Hence Riesz basis and frame appear as the tool to ensure the stability of sampling series. In this dissertation, first, we observe the sampling theorem for channeled sampling and oversampling in two channel sampling. Oversampling is to sample the input signal with rate greater than the Nyquist sampling rate. Oversampling is widely used in noise reduction, recovery of finite missing samples, and so on. Here we consider the recovery of finite missing samples in two-channel sampling or multi-channel sampling for band-pass or harmonic signal. Also we consider the sampling theory for MIMO system, we give a necessary and sufficient condition to have a stable sampling formula and consider the error estimate when we apply the sampling theory to signals which are not band-limited. Next, we consider the sampling on shift-invariant space. Although Shannon``s sampling theorem have made a big amount of influence on sampling theory, but it relies on the use of band-limited function, and the sinc function has very slow decay so that it makes a computation inefficient. Hence many people c...