| 姓名 |
古文仁(Wen-Jen Ku)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
歐氏空間富式分析的探討
(A study of Fourier series in Euclidean spaces)
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| 相關論文 | |
| 檔案 |
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| 摘要(中) |
在此篇論文裡，我們先探究在不同函數空間上的傅立葉轉換，例如說在L^1空間、在L^p空間1 < p ≤ 2及在Schwartz空間。接下來，我們會利用一些性質和定理去探究Hardy-Littlewood的極大函數，並證明其有weak (1,1)和strong(p,p)" 1 < p≤∞" 的性質。最後，我們將探討奇異積分算子的有界性，我們將專注在Hilbert transform。 |
| 摘要(英) |
In this thesis, we study various properties of Fourier transform. We first study the Fourier transform on Schwartz classes, and extend to L^p spaces for 1 ≤p≤2. Secondly, we shall focus on the Hardy-Littlewood maximal function, and prove that it is weak (1, 1) and strong(p, p) "for 1 < p≤∞" . At the end, we will discuss one of the most important singular intergrals, the so-called Hilbert transform. |
| 關鍵字(中) |
★ 傅利葉級數
★ 歐氏空間
★ 施瓦茨空间
★ 哈代－李特爾伍德極大函數
★ 奇異積分算子
★ 希爾伯特轉換 |
關鍵字(英) |
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| 論文目次 |
摘要 iii
Abstract vi
Introduction vii
Contents viii
1. Fourier Series and Intergrals 1
1-1 Fourier coefficients and series 1
1-2 Criteria for pointwise convergence 2
1-3 Fourier series of continuous functions 8
1-4 Convergence in norm 11
1-5 Summability method 11
1-6 The Fourier transform of L1 functions 14
1-7 The Schwartz class and tempered distributions 18
1-8 The Fourier transform on Lp,1 < p ≤ 2 22
2. The Hardy-Littlewood Maximal Function 26
2-1 Approximations of the identity 26
2-2 Weak-type inequalities and almost everywhere convergence 29
2-3 The Marcinkiewicz interpolation theorem 31
2-4 The Hardy-Littlewood maximal function 34
2-5 The dyadic maximal function 38
2-6 The weak (1, 1) inequality for the maximal function 41
3. The Hilbert Transform 45
3-1 The conjugate Poisson kernel 45
3-2 The principle value of 1/x 46
3-3 The theorems of M.Riesz and Kolmogorov 48
3-4 Truncated integrals and pointwise convergence 53
Reference 59 |
| 參考文獻 |
Fourier Analysis by Javier Duoandikoetxea. Translated and revised dy David Cruz-Uribe, SFO. Graduate Studies in Mathematics. Volume29. |
| 指導教授 |
沈俊嚴
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審核日期 |
2016-6-15 |
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