136,990 research outputs found

    An Analysis of the Theory of Functions of One Real Variable

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    Few undergraduates are aware that the Riemann integral taught in introductory calculus courses has only limited application-essentially this integral can be used only to integrate continuous functions over intervals. The necessity to integrate a broader class of functions over a wider range of sets that arises in many applications motivates the theory of abstract integration and functional analysis. The founder of this theory was the French mathematician Henri Lebesgue, who in 1902 defined the Lebesgue measure of subsets of the real line. The purpose of this project is to elucidate the theory of abstract measure spaces and of important spaces of functions (a critical example of which are Banach spaces), and extend the application of this theory. Developing the tools for doing so has been the focus of my advisor Professor Dmitry Khavinson and me over the past three years. The primary goal of the thesis is to make this highly formal and abstract material accessible to an undergraduate having only a year of coursework in advanced calculus. These concepts are typically introduced at the graduate level, but the ideas require only a familiarity with the analytic style of proof learned as an undergraduate. It would be advantageous to expose advanced undergraduates to this material since these ideas form the foundation for how mathematical research is done at the professional level. The addition of interesting and practical examples (which are scarce in the standard graduate texts) will help to make the concepts more familiar and down-to-earth. The motivation for a new theory of integration came from the Riemann integral\u27s apparent inability to operate on functions that fail to be continuous. For example, the Riemann integral of the function that assigns the value 1 to rational numbers and 0 to irrational numbers can be evaluated over the interval [0, 1] with equally valid justification to be 0 or 1. This is because the definition of the Riemann integral depends on partitioning the domain of the function to be integrated, and finding the maximum and minimum values of the function over each partition. The Lebesgue integral, on the other hand, partitions the range of the function to be integrated and then considers the length of the Jason Reed and Dmitry Khavinson pre-image of each partition as well as the maximum and minimum values of the function of the partition. The utility of this change of perspective arises when we refine what is meant by length in the aforementioned pre-image. The Riemann integral requires that the domain consist of intervals of real numbers (where length makes sense), while the Lebesgue integral can be used with a much broader class of sets. Lebesgue modified the notion of length by defining the measure of a set E to be the smallest possible total length of all collections of intervals that cover E. Using this ingenious method, Lebesgue constructed a theory of integration which forms the most useful example of all general integration theories. The theory has important applications in many areas of science and engineering as well as probability and statistics. Our approach to the subject has emphasized theory developed in H.L. Royden\u27s classic text, Real Analysis. My project has included analysis of each concept in the text, and I have developed for each major subject a collection of problems solved and applications of major theorems that were explored. The result has been comprehension of many of the foundational ideas in the field. We have used a number of supplemental texts to gain depth of understanding where Royden\u27s text provides only a survey, such as the Riesz Representation theorem, and to extend important ideas, such as the consideration of complex-valued (in addition to real-valued) measures. The synthesis has been a comprehensive paper which describes the theoretical directions the research has taken, the major results and theorems with proof, and applications and examples which are worked out in detail. The final record of my research will be divided into the following six sections: Lebesgue measure, Lebesgue integral, relationship between differentiation and Lebesgue integration, Banach space theory, abstract measure theory, and general integration theory. The analysis encompasses discussion of the main ideas (what it means for a set function to be a measure, how an integral can be defined in a coherent way with respect to a measure, when the derivative of an integral of a function is the function itself, different ideas about what it means for a sequence of functions to converge to a function, what are the properties of Banach spaces and why they are useful, etc.), as well as important ideas and theorems that interrelate these concepts (i.e., when we can interchange the limit of a sequence of functions and the integral, how we can represent a bounded linear functional, the structure of certain spaces of integrable functions)

    Order-type Henstock and McShane integrals in Banach lattice setting

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    We study Henstock-type integrals for functions defined in a compact metric space TT endowed with a regular σ\sigma-additive measure μ\mu, and taking values in a Banach lattice XX. In particular, the space [0,1][0,1] with the usual Lebesgue measure is considered.Comment: 5 page

    A note on set-valued Henstock--McShane integral in Banach (lattice) space setting

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    We study Henstock-type integrals for functions defined in a Radon measure space and taking values in a Banach lattice XX. Both the single-valued case and the multivalued one are considered (in the last case mainly cwk(X)cwk(X)-valued mappings are discussed). The main tool to handle the multivalued case is a R{\aa}dstr\"{o}m-type embedding theorem established in [50]: in this way we reduce the norm-integral to that of a single-valued function taking values in an MM-space and we easily obtain new proofs for some decomposition results recently stated in [33,36], based on the existence of integrable selections. Also the order-type integral has been studied: for the single-valued case some basic results from [21] have been recalled, enlightning the differences with the norm-type integral, specially in the case of LL-space-valued functions; as to multivalued mappings, a previous definition ([6]) is restated in an equivalent way, some selection theorems are obtained, a comparison with the Aumann integral is given, and decompositions of the previous type are deduced also in this setting. Finally, some existence results are also obtained, for functions defined in the real interval [0,1][0,1].Comment: This work has been modified both as regards the drawing that with regard to the assumptions. A new version is contained in the paper arXiv:1503.0828

    On the Henstock-Kurzweil Integral for Riesz-space-valued Functions on Time Scales

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    We introduce and investigate the Henstock-Kurzweil (HK) integral for Riesz-space-valued functions on time scales. Some basic properties of the HK delta integral for Riesz-space-valued functions are proved. Further, we prove uniform and monotone convergence theorems.Comment: This is a preprint of a paper whose final and definite form is with 'J. Nonlinear Sci. Appl.', ISSN 2008-1898 (Print) ISSN 2008-1901 (Online). Article Submitted 17-Jan-2017; Revised 17-Apr-2017; Accepted for publication 19-Apr-2017. See [http://www.tjnsa.com

    Stochastic evolution equations driven by Liouville fractional Brownian motion

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    Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of L(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motions (fBm) with arbitrary Hurst parameter in the interval (0,1). For Hurst parameters in (0,1/2) we show that a function F:(0,T)\to L(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fBm if and only if it is stochastically integrable with respect to an H-cylindrical fBm with the same Hurst parameter. As an application we show that second-order parabolic SPDEs on bounded domains in \mathbb{R}^d, driven by space-time noise which is white in space and Liouville fractional in time with Hurst parameter in (d/4,1) admit mild solution which are H\"older continuous both and space.Comment: To appear in Czech. Math.

    The vector-valued tent spaces T^1 and T^\infty

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    Tent spaces of vector-valued functions were recently studied by Hyt\"onen, van Neerven and Portal with an eye on applications to H^\infty-functional calculi. This paper extends their results to the endpoint cases p = 1 and p = \infty along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case p = 1, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.Comment: 19 pages, minor corrections mad
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