Non-linear Approximation For Sinc Function And Their Applications

Shannon sampling theorem is the basic frame for signal communication and image processing. Based on the sampling formula from the Shannon sampling theorem, the bandlimited signal can be exactly recovered without loss. The sinc function can be considered as the interpolation kernel in the sampling formula and it is an ideal low pass filter. However, people tend to use only finite terms in the sum of the sampling formula when recovering the signal, which will lead to the truncated error. If we want to get a suitable truncation error, then a lot of terms in the sum are required, which brings a great computation. In practice, signals are never exactly bandlimited, thus there are no adequate reasons for the sinc function as the ideal interpolation kernel. Therefore, people modified the Shannon sampling formula from two aspects. On one hand, constructing a suitable function and adding this function into the sum of finite terms in the Shannon sampling formula, which will reduce the truncation error. This constructed function is considered as the convergence factor. On other hand, Constructing a limited supported function and it need to satisfy some basic properties of the sinc function. This limited supported function will replace the sinc function in the sampling formula.In the above two aspects, we will study these approximation problems concerning sinc func-tion. Moveover, we will prove again that the linear multistep method is instable, when it achieves the highest approximation order. The main results of this dissertation can be summarized as fol-lows.1. In Chapter 1, some related researches about the sinc function, spline function, Pade and algebraic function approximation are introduced.2. In Chapter 2, by studying Pade approximation of the sinc function, We give the [2/4]-Pade approximation of the sinc function. Then, the [2/4]-Pade approximation can be consid-cred as the convergence factor and is added into Shannon sampling formula. Finally, by comparing it with other two convergence factors in numerical experiments, the [2/4]-Pade approximation can also achieve very accurate results.3. In Chapter 3, [2/6], [0/2], [0/4] and [0/6]-Pade approximation of sinc function are given and they are be considered as the convergence factors respectively. By comparison between [2/6]-Pad6 approximation, other three kinds of Pade approximation and three kinds of con- vergence factors in the previous section, the [2/4]-Pade approximation can also achieve very accurate results.4. In Chapter 4, based on the research about the cubic/linear rational spline, we study the the cubic/linear rational spline approximation of the sinc function and get a kind of the cubic/linear rational spline with one free parameter. By an analysis of this cubic/linear rational spline in frequency domain, we obtain that when the value of parameter is two, the cubic/linear rational spline has flat spectrum at low frequencies. Some other reasonable choices of the parameter are given. Finally, by comparing with other methods in image processing, our method can also achieve satisfied results.5. In Chapter 5, based on the research about the algebraic function approximation to the expo-nential function, we study the [1,n] algebraic function approximation of exponential func-tion. Moreover, we again prove the linear multistep method is instable, when it achieves the highest approximation order.