Large deviation theory has experienced much development and interest in the last two decades. A large deviation principle is the exponential decay of the probability of increasingly rare events and the computation of a rate or entropy function which measures the rate of decay. Within the probability literature there has been much use made of these rates in diverse applications. These large deviation principles have been discovered for independent and identically distributed random variables, as well as random vectors and these have been extended to some cases of weak dependence. In this thesis we prove large deviation principles for finite dimensional distributions of scaling limits of random measures. Functional approaches to large deviation theory using test functions as dual objects to random measures are also developed. These results are applied to some important classes of models, in particular Poisson point processes, Poisson center cluster processes and doubly stochastic point processes. / Graduation date: 1991

This dissertation concerns two topics in analysis. The rst section is an exposition of the Henstock-Kurzweil integral leading to a necessary and su cient condition for the change of variables formula to hold, with implications for the change of variables formula for the Lebesgue integral. As a corollary, a necessary and suf- cient condition for the Fundamental Theorem of Calculus to hold for the HK integral is obtained. The second section concerns a challenge raised in a paper by O. Lazarev and E. H. Lieb, where they proved that, given f1….,fn ∈ L1 ([0,1] ; C), there exists a smooth function φ that takes values on the unit circle and annihilates span {f1...., fn}. We give an alternative proof of that fact that also shows the W1,1 norm of φ can be bounded by 5πn + 1. Answering a question raised by Lazarev and Lieb, we show that if p > 1 then there is no bound for the W1,p norm of any such multiplier in terms of the norms of f1...., fn. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The thesis is divided into three parts. Part One, Measure and Measure Spaces, contains the introduction; and in later chapters this part approaches measure theory by the most natural route. It begins in a detailed fashion by developing the theory of set length through several levels of generalization, including the concepts of ordinary interval length, Jordan content, and Lebesgue measure. Then, without details, it discusses the extension of this theory of measure to more complex Euclidean sets and, finally, to abstract sets. Part Two, The Integral on Measure Spaces, introduces the theory of Lebesgue integration. The integrals treated gere are the measure dependent Lebesgue and general Lebesgue integrals. Chapter 6 in this part establishes a link between the Riemann integral and Jordan content, described in Chapter 3 of Part One, and between the Lebesgue integral and Lebesgue measure, described in Chapter 4 of Part One. Chapter 7 establishes an analogous link between the general Lebesgue integral and general measure spaces introduced in Chapter 5 of Part One. Chapter 8 , Measure on a Semialgebra, stands as a bridge between the historic and abstract versions of the Lebesgue Theory; for it generalizes the procedure used to construct the geometric integral, and shows that this procedure will produce general integrals as well. Part Three, A More General Integral , considers the theory of the Daniell integral. This integral is defined in more general terms than the measure-dependent Lebesgue integral of Part Two. Nevertheless, a harmonious resolution is reached in the theorems that show how the lines supporting the two theories can be drawn together / 2031-01-01

在這篇論文中，我們將會首先討論什麼概率測度μ 上的L²空間會存在指數型正交基(exponential orthonormal basis) ，而μ 則稱為一個譜測度若指數型正交基存在。這個問題源於1974年的Fuglede猜想和Jorgensen與Pedersen對分形譜測度存在性的研究。我們有興趣理解怎麼樣的測度會是譜測度，而對於沒有指數型正交基的測度，我們則探討它們會否存在更廣義且在Fourier分析中常用的指數型基，如Riesz基或Fourier框架(Fourier frame) 。 / 我們知道一個測度可以唯一分解成離散、奇異和絶對連續三部份。我們首先証明譜測度肯定是純型(pure type) 。若測度是絶對連續，我們對有Fourier框架的測度的密度給出一個完全的刻直到。這個結果對研究Gabor框架的問題都有幫助。對於離散且只有有限個非零質量原子的測度，我們証明它們全都都有Riesz基。在最困難作出一般討論的奇異測度中，我們透過譜測度與離散測度的卷積找出了有Riesz基但沒有指數型正交基的奇異測度。我們進而探討自彷函數送代系統(affine IFSs) ，我們証明到如果一個自彷函數送代系統是測度分離且有Fourier框架，那麼它在每一個函數的概率權都是一樣的。我們亦証明了Laba-Wang猜想在絶對連續的自相似測度上是正確的。這些結果都表示了一個有Fourier框架的測度都應該在其支撐上有一定的均勻性。 / 在論文的第二部分我們會探討自彷tile其Fourier變換的零點集。在自彷tile的研究中，其中一個基本問題就是刻劃其數字集(digit set)使得那自彷函數送代系統的不變集能以平移密鋪空間。透過Kenyon條件，我們可將這個問題轉化成理解Fourier變換的零點集。男一方面，指數型正交基的存在性亦需要我們探討Fourier變換的零點集，而自彷tile 的Fourier變換是可以明確寫出來的。這使自彷tile成為一個很好去研究tilings和譜測度相互關係的好例子。 / 我們利用了分圓多項式(cyclotomic polynomials)對一維自彷tile的零點集進行了一個詳細的研究。從這裡我們把tile的數字集寫成某些分圓多項式的乘積。這個乘積亦可以一個樹上的切集(blocking)表示出來。這個表示亦把tile數字集的乘積形式(product-forms) 一般化成高階乘積形式。我們証明了在任何維數的tile數字集都是整數tile(即它們能平移密鋪整數集Z) 。這個結果讓我們能使用Coven和Meyerowitz所提出的整數tile分解方法，來使tile數字集完整刻劃成高階模乘積形式(high order modulo product-forms) 當數字集的數目為p[superscript α]q而p，q則是質數。由於我們對零點集亦完全清楚，這對在自彷tile上尋找完備的指數型正交基提供了一個新的方向。 / In this thesis, we will first consider when a probability measure μ admits an exponential orthonormal basis on its L² space (μ is called spectral measures).This problem originates from the conjecture of Fuglede in 1974, and the discovery of Jorgensen and Pedersen that some fractal measures also admit exponential orthonormal bases, but some do not. It generates a lot of interest in understand- ing what kind of measures are spectral measures. For those measures failing to have exponential orthonormal bases, it is interesting to know whether such mea- sures still have Riesz bases and Fourier frames, which are generalized concepts of orthonormal bases with wide range of uses in Fourier analysis. / It is well-known that a measure has a unique decomposition as the discrete, singular and absolutely continuous parts. We first show that spectral measures must be of pure type. If the measure is absolutely continuous, we completely classify the class of densities of the measures with Fourier frames. This result has new applications to topics in applied harmonic analysis, like the Gabor analysis. For the discrete measures with finite number of atoms, we show that they all have Riesz bases. For the case of singular measure, which is the most difficult one, we show that there exist measures with Riesz bases but not orthonormal bases by considering convolution between spectral measures and discrete measures. We then investigate affine iterated function systems (IFSs), we show that if an IFS has measure disjoint condition and admits a Fourier frame, then the probability weights are all equal. Moreover, we also show that the Łaba-Wang conjecture is true if the self-similar measure is absolutely continuous. These results indicate that measures with Fourier frames must have certain kind of uniformity on the support. / In the second part of the thesis we study the zero sets of Fourier transform of self-affine tiles. One of the fundamental problems in self-affine tiles is to classify the digit sets so that the attractors form tiles. This problem can be turned to study the zeros of the Fourier transform via the Kenyon criterion. On the other hand, existence of exponential orthonormal bases requires us to know the zero sets of the Fourier transform. Self-affine tiles are translational tiles arising from IFSs with its Fourier transform written explicitly. It therefore serves as an ideal place to investigate the relation of tilings and spectral measures. / We carry out a detail study in the zero sets of the one-dimensional tiles using cyclotomic polynomials. From this we characterize the tile digit sets through some product of cyclotomic polynomials represented in terms of a blocking in a tree, which is a generalization of the product-form to higher order. We show that tile digit sets in any dimension are integer tiles. This result allows us to use the decomposition method of integer tiles by Coven and Meyerowitz to provide the explicit classification of the tile digit sets in terms of the higher order modulo product-forms when number of the digits is p[superscript α]q, p, q are primes. Since the zero sets are completely known, this provides us a new approach to study the existence of complete orthogonal exponentials in the self-affine tiles on R¹. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Lai, Chun Kit. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 128-135). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.9 / Chapter 1.1 --- Background and Motivations --- p.9 / Chapter 1.2 --- Results on spectral measures --- p.13 / Chapter 1.3 --- Results on self-affine tiles --- p.16 / Chapter 2 --- Fourier Frames: Absolutely Continuous Measures --- p.21 / Chapter 2.1 --- Beurling densities --- p.22 / Chapter 2.2 --- Law of pure type --- p.28 / Chapter 2.3 --- Absolutely continuous F-spectral measures --- p.31 / Chapter 2.3.1 --- Proof by Beurling densities --- p.32 / Chapter 2.3.2 --- Proof by translational absolute continuity --- p.35 / Chapter 2.4 --- Applications to applied harmonic analysis --- p.40 / Chapter 2.5 --- Remarks and open questions --- p.42 / Chapter 3 --- Fourier Frames: Discrete and Singular Measures --- p.45 / Chapter 3.1 --- Discrete measures --- p.46 / Chapter 3.2 --- Convolutions with discrete measures --- p.50 / Chapter 3.3 --- Self-affine measures --- p.56 / Chapter 3.4 --- Iterated function systems on R¹ --- p.65 / Chapter 3.5 --- Concluding remarks --- p.70 / Chapter 4 --- Spectral structure of tile digit sets --- p.74 / Chapter 4.1 --- Preliminaries --- p.76 / Chapter 4.2 --- Modulo product-forms --- p.81 / Chapter 4.3 --- Higher order product-forms --- p.86 / Chapter 4.4 --- Φ-tree, blocking and kernel polynomials --- p.90 / Chapter 5 --- Classifications of tile digit sets --- p.101 / Chapter 5.1 --- Tile digit sets --- p.101 / Chapter 5.2 --- The p[superscript α]q[superscript β] integer tiles --- p.105 / Chapter 5.3 --- Tile digit sets for b = p[superscript α]q --- p.112 / Chapter 5.4 --- Self-similar measures: Absolute continuity --- p.122 / Chapter 5.5 --- Remarks --- p.126

by Li Jian Lin. / "November 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 85-90) / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

by Lo Yiu Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-90). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1. --- Main Idea --- p.5 / Chapter 1.2. --- Organization --- p.8 / Chapter Chapter 2. --- Geometric and Analytic Preliminaries --- p.9 / Chapter 2.1. --- A Review on Measure Theory --- p.9 / Chapter 2.2. --- Submanifolds in Rn --- p.11 / Chapter 2.3. --- Several Results from PDEs --- p.17 / Chapter 2.4. --- Biharmonic Comparison Lemma --- p.20 / Chapter Chapter 3. --- Approximate Graphical Decomposition --- p.24 / Chapter 3.1. --- Some Preliminaries --- p.24 / Chapter 3.2. --- Approximate Graphical Decomposition --- p.30 / Chapter Chapter 4. --- Existence & Regularity of Measure-theoretic Limits of Minimizing Sequence --- p.41 / Chapter 4.1. --- Willmore Functional and Area --- p.41 / Chapter 4.2. --- Existence of Measure-theoretic Limit of Minimizing Sequence --- p.45 / Chapter 4.3. --- Higher Regularity at Good Points --- p.54 / Chapter 4.4. --- Convergence in Hausdorff Distance Sense --- p.62 / Chapter 4.5. --- Regularity near Bad Points --- p.64 / Chapter Chapter 5. --- Existence of Genus 1 Minimizers in Rn --- p.83 / References --- p.89

Lai, Chun Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 83-87). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Spectral sets in Rd --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Fundamental domains and convex sets in Rd --- p.15 / Chapter 2.3 --- Finite union of cubes --- p.20 / Chapter 3 --- Spectral theory on discrete groups --- p.27 / Chapter 3.1 --- Finite groups and Zd --- p.28 / Chapter 3.2 --- Rational spectrums and tiling sets --- p.32 / Chapter 3.3 --- Fuglede´ةs Problem in R1 --- p.37 / Chapter 3.4 --- "Failure of Fuglede´ةs Conjecture in Rd, d >3" --- p.42 / Chapter 4 --- Self-similar tiles in R1 --- p.49 / Chapter 4.1 --- Basics of self-similar tiles --- p.49 / Chapter 4.2 --- Self-similar tile digit sets and spectral problem --- p.52 / Chapter 4.3 --- Kenyon criterion --- p.55 / Chapter 5 --- Spectral self-similar measures --- p.66 / Chapter 5.1 --- Spectral self-similar measures --- p.66 / Chapter 5.2 --- One-dimensional self-similar measures --- p.72 / Chapter 5.3 --- General properties of spectral measures --- p.80 / Bibliography --- p.83

Vincent W. S. Chan. / Prepard under NASA Grant NGL-22-009-013. Originally presented as the author's thesis (Ph.D), M.I.T. Dept. of Electrical Engineering, l974. / Bibliography: p.128-129.

Robert E. Wernikoff. / "January 11, 1957." / Bibliography: p. 73. / Army Signal Corps Contract DA36-039-sc-64637. Dept. of the Army Task 3-99-06-108 and Project 3-99-00-100.

The investigation of this paper is introduced by describing an important collection of classes arising from a class which forms the domain of an abstract measure. Characterizations of the different types of classes will be formulated in order to facilitate the construction and the identification of the various classes considered.