【摘要】：在對(duì)標(biāo)量和四元素值L~2-函數(shù)的實(shí)Paley-Wiener定理(以下簡稱QFT)的研究總結(jié)后,本文推廣并證明了 R~2函數(shù)上四元素值Schwartz函數(shù)和Lp-函數(shù)的四元素傅里葉變換的實(shí)Paley-Wiener定理。首先,在經(jīng)典傅里葉變換下,基于反演定理的基本方法,我們系統(tǒng)地研究了Rd上Schwartz函數(shù),Lp-函數(shù)和分布下傅里葉變換的實(shí)Paley-Wiener定理。作為一個(gè)應(yīng)用,我們展示了如何通過不涉及域移位的方法實(shí)現(xiàn)經(jīng)典的Paley-Wiener定理的證明。我們首先針對(duì)Schwartz函數(shù)來展開研究其實(shí)Paley-Wiener定理。其次,我們對(duì)QFT在四元素領(lǐng)域方面進(jìn)行了相關(guān)研究,并且給出了QFT在實(shí)際應(yīng)用中的一系列相關(guān)性質(zhì)。不同形式的QFT會(huì)使我們推導(dǎo)出不同的Plancherel定理和Parseval定理,這些定理在后面的定理證明中發(fā)揮了重要的作用。與經(jīng)典傅里葉變換的實(shí)Paley-Wiener定理相比,四元素傅里葉變換在f∈F2(R~2 H) 上的實(shí)Paley-Wiener可以更直觀地表示為:通過在IR~2上偏導(dǎo)數(shù)的范式來描述四元素傅里葉變換具有緊致集。最后,經(jīng)典的Paley-Wiener定理描述了L~2空間上函數(shù)的傅里葉變換。這個(gè)函數(shù)是L~2空間上指數(shù)型的整函數(shù),并且其支集支在一個(gè)有限的對(duì)稱區(qū)間上。經(jīng)典的Paley-Wiener定理在各種變換中得到了廣泛_的應(yīng)用。最近,通過B ang的實(shí)Paley-Wiener定理和Fu的文章的學(xué)習(xí),我們想到把L~2 (R~2; H)上的Paley-Wiener定理擴(kuò)展到Lp(R~2;H)空間上。這里首先遇到了一個(gè)問題,當(dāng)p≠ 2時(shí),沒有關(guān)于QFT的Plancherel定理。這里我們用de Jeu文章中Lp(Rd;空間中Paley-Wiener定理的方法,即利用關(guān)于Lp核逐點(diǎn)逼近思想。另外四元素H的非交換性,我們就不能直接把經(jīng)典傅里葉變換上的卷積運(yùn)算推廣到四元素域。為了克服這個(gè)困難,我們利用調(diào)和分析中恒等逼近思想。通過對(duì)之前L~2空間中四元素傅里葉變換實(shí)Paley-Wiener定理的研究,我們證明了實(shí)Paley-Wiener定理對(duì)于四元素在Lp空間上也是成立的。
[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，

本文編號(hào)：2427595