The Experts below are selected from a list of 87 Experts worldwide ranked by ideXlab platform
John W Hagood - One of the best experts on this subject based on the ideXlab platform.
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the lebesgue Differentiation Theorem via nonoverlapping interval covers
Real analysis exchange, 2004Co-Authors: John W HagoodAbstract:A short proof is given for the Lebesgue Differentiation Theorem using a variation of the Heine-Borel covering property, without reliance on sophisticated approaches such as Vitali covers and the rising sun lemma.
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the lebesgue Differentiation Theorem via nonoverlapping interval covers
Real analysis exchange, 2004Co-Authors: John W HagoodAbstract:A short proof is given for the Lebesgue Differentiation Theorem using a variation of the Heine-Borel covering property, without reliance on sophisticated approaches such as Vitali covers and the rising sun lemma. In this paper we use a variation of the Heine-Borel covering property to prove the Theorem due to Lebesgue that every monotone function f : [a, b]→ R is differentiable almost everywhere. The approach is more accessible than typical treatments that use Vitali covers, the rising sun lemma or other methods [1, 2, 3, 4, 5]. Throughout λ represents Lebesgue measure on the real line. A family of nondegenerate compact intervals C is a right adapted interval cover of a set E ⊆ R if for each x ∈ E there is an interval [L(x), R(x)] ∈ C such that L(x) < x < R(x) and [s,R(x)] ∈ C for all s ∈ [L(x), x]. The term left adapted interval cover is defined similarly, and we refer to either of these as an adapted interval cover. We say that a family of compact intervals is nonoverlapping if the interiors of the intervals are pairwise disjoint. Covering Lemma. If C is an adapted interval cover of a compact set K ⊆ R, then there is a finite collection of nonoverlapping intervals in C that covers K. Proof. Without loss of generality, suppose that C is right adapted. Let a = minK and b = maxK and let A be the set of all t ∈ [a, b] such that C has a finite nonoverlapping subcover of [a, t] ∩K. Then a ∈ A, so A is nonempty. Let β = supA. We first show that β ∈ K. Otherwise, β lies in a component (c, d) of [a, b]\K and there is a finite collection D of nonoverlapping intervals in C that covers [a, c] ∩ K. Then D can be modified by deleting extraneous
Donald M. Stull - One of the best experts on this subject based on the ideXlab platform.
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Polynomial Space Randomness in Analysis with Application to the Lebesgue Differentiation Theorem.
arXiv: Computational Complexity, 2015Co-Authors: Xiang Huang, Donald M. StullAbstract:We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue Differentiation Theorem. We generalize Ko's framework for polynomial space computability in $\mathbb{R}^n$ to define \textit{weakly pspace-random} points, a new variant of polynomial space randomness. We show that the Lebesgue Differentiation Theorem holds for every weakly pspace-random point.
Slavíková Lenka - One of the best experts on this subject based on the ideXlab platform.
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Weighted inequalities and properties of operators and embeddings on function spaces
2016Co-Authors: Slavíková LenkaAbstract:The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue Differentiation Theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this Theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the..
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Weighted inequalities and properties of operators and embeddings on function spaces
Univerzita Karlova Matematicko-fyzikální fakulta, 2016Co-Authors: Slavíková LenkaAbstract:Tato disertační práce je věnována studiu nejrůznějších vlastností Banachových prostorů funkcí se zvláštním zřetelem k aplikacím v teorii Sobolevových prostorů a v harmonické analýze. Práce sestává ze čtyř článků. V prvním z nich zkoumá- me vnoření vyššího řádu prostorů Sobolevova typu vybudovaných nad Bana- chovými prostory funkcí s normou invariantní vůči nerostoucímu přerovnání. Mimo jiné ukážeme, že optimální Sobolevova vnoření vyššího řádu plynou z izoperimetrických nerovností. Ve druhém článku se zabýváme otázkou, kdy je výše zmíněný prostor Sobolevova typu Banachovou algebrou vzhledem k bodové- mu násobení funkcí. Dokážeme, že vnoření Sobolevova prostoru do prostoru esen- ciálně omezených funkcí je odpovědí na tuto otázku v mnoha standardních i ne- standardních případech. Třetí článek je věnován problému platnosti Lebesgueovy věty o derivování v kontextu Banachových prostorů funkcí s normou invariantní vůči nerostoucímu přerovnání. Nalezneme nutnou a postačující podmínku pro platnost této věty vyjádřenou pomocí konkavity jistého funkcionálu závisejícího na dané normě a poskytneme rovněž několik alternativních charakterizací zada- ných pomocí vlastností...The present thesis is devoted to the study of various properties of Banach func- tion spaces, with a particular emphasis on applications in the theory of Sobolev spaces and in harmonic analysis. The thesis consists of four papers. In the first one we investigate higher-order embeddings of Sobolev-type spaces built upon rearrangement-invariant Banach function spaces. In particular, we show that optimal higher-order Sobolev embeddings follow from isoperimetric inequal- ities. In the second paper we focus on the question when the above-mentioned Sobolev-type space is a Banach algebra with respect to a pointwise multiplica- tion of functions. An embedding of the Sobolev space into the space of essentially bounded functions is proved to be the answer to this question in several standard as well as nonstandard situations. The third paper is devoted to the problem of validity of the Lebesgue Differentiation Theorem in the context of rearrangement- invariant Banach function spaces. We provide a necessary and sufficient condition for the validity of this Theorem given in terms of concavity of certain functional depending on the norm in question and we find also alternative characterizations expressed in terms of properties of a maximal operator related to the norm. The object of the final paper is the boundedness of the...Department of Mathematical AnalysisKatedra matematické analýzyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic
Xiang Huang - One of the best experts on this subject based on the ideXlab platform.
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Polynomial Space Randomness in Analysis with Application to the Lebesgue Differentiation Theorem.
arXiv: Computational Complexity, 2015Co-Authors: Xiang Huang, Donald M. StullAbstract:We study the interaction between polynomial space randomness and a fundamental result of analysis, the Lebesgue Differentiation Theorem. We generalize Ko's framework for polynomial space computability in $\mathbb{R}^n$ to define \textit{weakly pspace-random} points, a new variant of polynomial space randomness. We show that the Lebesgue Differentiation Theorem holds for every weakly pspace-random point.
Brian S Thomson - One of the best experts on this subject based on the ideXlab platform.
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vitali coverings and lebesgue s Differentiation Theorem
Real analysis exchange, 2004Co-Authors: Brian S ThomsonAbstract:The standard techniques used to prove the Lebesgue dierentiation Theorem (that monotonic functions are a.e. dierentiable) are presented in an unusual way that reveals more about their nature and allows greater generality.