四元素傅里叶变换相关问题的研究
发布时间：2019-02-21 14:48
【摘要】：在对标量和四元素值L~2-函数的实Paley-Wiener定理(以下简称QFT)的研究总结后,本文推广并证明了 R~2函数上四元素值Schwartz函数和Lp-函数的四元素傅里叶变换的实Paley-Wiener定理。首先,在经典傅里叶变换下,基于反演定理的基本方法,我们系统地研究了Rd上Schwartz函数,Lp-函数和分布下傅里叶变换的实Paley-Wiener定理。作为一个应用,我们展示了如何通过不涉及域移位的方法实现经典的Paley-Wiener定理的证明。我们首先针对Schwartz函数来展开研究其实Paley-Wiener定理。其次,我们对QFT在四元素领域方面进行了相关研究,并且给出了QFT在实际应用中的一系列相关性质。不同形式的QFT会使我们推导出不同的Plancherel定理和Parseval定理,这些定理在后面的定理证明中发挥了重要的作用。与经典傅里叶变换的实Paley-Wiener定理相比,四元素傅里叶变换在f∈F2(R~2 H) 上的实Paley-Wiener可以更直观地表示为:通过在IR~2上偏导数的范式来描述四元素傅里叶变换具有紧致集。最后,经典的Paley-Wiener定理描述了L~2空间上函数的傅里叶变换。这个函数是L~2空间上指数型的整函数,并且其支集支在一个有限的对称区间上。经典的Paley-Wiener定理在各种变换中得到了广泛_的应用。最近,通过B ang的实Paley-Wiener定理和Fu的文章的学习,我们想到把L~2 (R~2; H)上的Paley-Wiener定理扩展到Lp(R~2;H)空间上。这里首先遇到了一个问题,当p≠ 2时,没有关于QFT的Plancherel定理。这里我们用de Jeu文章中Lp(Rd;空间中Paley-Wiener定理的方法,即利用关于Lp核逐点逼近思想。另外四元素H的非交换性,我们就不能直接把经典傅里叶变换上的卷积运算推广到四元素域。为了克服这个困难,我们利用调和分析中恒等逼近思想。通过对之前L~2空间中四元素傅里叶变换实Paley-Wiener定理的研究,我们证明了实Paley-Wiener定理对于四元素在Lp空间上也是成立的。
[Abstract]:In this paper, we generalize and prove the real Paley-Wiener theorem of four-element valued Schwartz function and Lp- function on the R2 function after the research on the real Paley-Wiener theorem of scalar and four-element valued Ln 2- function (hereinafter referred to as QFT). Firstly, based on the basic method of inversion theorem, we systematically study the real Paley-Wiener theorem of Schwartz function, Lp- function and Fourier transform on Rd under classical Fourier transform. As an application, we show how to prove the classical Paley-Wiener theorem by using the method without domain shift. We first study the Paley-Wiener theorem for Schwartz functions. Secondly, we study QFT in the field of four elements, and give a series of related properties of QFT in practical application. Different forms of QFT will lead us to derive different Plancherel theorems and Parseval theorems, which play an important role in the later proofs of the theorems. Compared with the real Paley-Wiener theorem of classical Fourier transform, The real Paley-Wiener of four-element Fourier transform on f 鈭，
本文编号：2427595
[Abstract]:In this paper, we generalize and prove the real Paley-Wiener theorem of four-element valued Schwartz function and Lp- function on the R2 function after the research on the real Paley-Wiener theorem of scalar and four-element valued Ln 2- function (hereinafter referred to as QFT). Firstly, based on the basic method of inversion theorem, we systematically study the real Paley-Wiener theorem of Schwartz function, Lp- function and Fourier transform on Rd under classical Fourier transform. As an application, we show how to prove the classical Paley-Wiener theorem by using the method without domain shift. We first study the Paley-Wiener theorem for Schwartz functions. Secondly, we study QFT in the field of four elements, and give a series of related properties of QFT in practical application. Different forms of QFT will lead us to derive different Plancherel theorems and Parseval theorems, which play an important role in the later proofs of the theorems. Compared with the real Paley-Wiener theorem of classical Fourier transform, The real Paley-Wiener of four-element Fourier transform on f 鈭，
本文编号：2427595
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