【摘要】：Shannon采樣定理為信號(hào)通信和圖像處理奠定了嚴(yán)格的理論基礎(chǔ).根據(jù)Shannon采樣公式,有限帶寬信號(hào)可以被精確的恢復(fù).Sinc函數(shù)是Shannon采樣公式中的插值核.同時(shí)Sinc函數(shù)還被看作是一個(gè)理想的低通濾波器.在信號(hào)的實(shí)際恢復(fù)過(guò)程中,通常只涉及到Shannon采樣公式中的有限項(xiàng)求和,因此就會(huì)產(chǎn)生一個(gè)截?cái)嗾`差.如果要得到一個(gè)合適的截?cái)嗾`差,就需要很多項(xiàng)求和,因而就帶來(lái)了很大的計(jì)算量.另外,大多數(shù)信號(hào)都不是嚴(yán)格意義上的有限帶寬信號(hào),此時(shí)若仍把Sinc函數(shù)看作是理想的插值核,則缺乏一個(gè)合理的解釋.為了解決這些問(wèn)題,人們便開(kāi)始從兩方面對(duì)Shannon采樣公式的有限項(xiàng)求和進(jìn)行改進(jìn).一方面,構(gòu)造一個(gè)合適的函數(shù)將其加入到Shannon采樣公式的有限項(xiàng)求和中,來(lái)減小截?cái)嗾`差,此時(shí)構(gòu)造的函數(shù)被稱為收斂因子；另一方面,構(gòu)造一個(gè)具有緊支集的函數(shù),同時(shí)該函數(shù)需要滿足Sinc函數(shù)的一些性質(zhì).最后在S hannon采樣公式的有限項(xiàng)求和中,用構(gòu)造的函數(shù)來(lái)代替Sinc函數(shù).本文將從這兩方面來(lái)考慮Sinc函數(shù)的逼近問(wèn)題.另外,我們將再次論證當(dāng)線性多步法達(dá)到最高逼近階時(shí),該差分格式是不穩(wěn)定的.本文分為五章,具體安排如下：1.第一章,我們介紹了Sinc函數(shù)、樣條函數(shù)、Pade逼近和代數(shù)函數(shù)逼近的相關(guān)內(nèi)容及研究情況.2.第二章,通過(guò)研究Sinc函數(shù)的Pade逼近,我們給出了Sinc函數(shù)的[2/4]型Pade逼近.然后把[2/4]型Pade逼近看作是一個(gè)收斂因子,將其加入到Shannon采樣公式的有限項(xiàng)求和中.最后和已有的收斂因子進(jìn)行了數(shù)值實(shí)驗(yàn)比較,將[2/4]型Pade逼近作為收斂因子的有限項(xiàng)求和也能得到很好的精度.3.第三章,我們給出了Sinc函數(shù)的[2/6]型、[0/2]型、[0/4]型和[0/6]型Pade逼近.然后將[2/6]型Pade逼近和另外三類Pade逼近以及第二章中的三類收斂因子進(jìn)行數(shù)值實(shí)驗(yàn)比較,[2/6]型Pade逼近作為收斂因子能得到很好的精度.4.第四章,基于3/1型有理樣條函數(shù)已有的研究,我們研究了Sinc函數(shù)的3/1型有理樣條函數(shù)逼近,并得到了一類含參數(shù)的3/1型有理樣條函數(shù).通過(guò)分析它的頻譜在原點(diǎn)處的泰勒展開(kāi)式,我們得到：當(dāng)參數(shù)值取2時(shí),該3/1型有理樣條函數(shù)在低頻處有平坦譜.另外,還給出了參數(shù)的其它幾種合理的取值.最后與已有的幾種方法通過(guò)圖像處理進(jìn)行比較,我們的方法也能得到很好的圖像處理效果.5.第五章,我們從指數(shù)函數(shù)的代數(shù)函數(shù)逼近角度,研究了指數(shù)函數(shù)的[1,n]級(jí)代數(shù)函數(shù)逼近以及與線性多步法的聯(lián)系.最后我們給出了一個(gè)新的證明：當(dāng)線性多步法達(dá)到最高逼近階時(shí),其差分格式是不穩(wěn)定的.
[Abstract]:Shannon sampling theorem lays a strict theoretical foundation for signal communication and image processing. According to the Shannon sampling formula, the finite bandwidth signal can be accurately recovered. Inc function is the interpolation kernel in the Shannon sampling formula. At the same time, Sinc function is also regarded as an ideal low-pass filter. In the actual recovery process of the signal, only the finite term summation in the Shannon sampling formula is usually involved, so a truncation error will be produced. If we want to get a suitable truncation error, we need a lot of items to sum up, so it brings a lot of computation. In addition, most of the signals are not limited bandwidth signals in the strict sense, and if the Sinc function is still regarded as an ideal interpolation kernel, there is a lack of a reasonable explanation. In order to solve these problems, people began to improve the finite term sum of Shannon sampling formula from two aspects. On the one hand, a suitable function is constructed to add it to the finite term sum of Shannon sampling formula to reduce the truncation error. At this time, the constructed function is called convergence factor. On the other hand, a function with compact support set is constructed, and the function needs to satisfy some properties of Sinc function. Finally, in the summation of finite terms of S hannon sampling formula, the constructed function is used instead of Sinc function. In this paper, we will consider the approximation of Sinc functions from these two aspects. In addition, we will prove once again that the difference scheme is unstable when the linear multistep method reaches the highest approximation order. This paper is divided into five chapters, the specific arrangements are as follows: 1. In the first chapter, we introduce the related contents and research of Sinc function, Spline function, Pade approximation and Algebra function approximation. 2. In chapter 2, by studying the Pade approximation of Sinc function, we give the [2 鈮，

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