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Analysis in Integer and Fractional Dimensions.

By: Series: Cambridge Studies in Advanced MathematicsPublisher: Cambridge : Cambridge University Press, 2001Copyright date: ©2001Description: 1 online resource (578 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780511153075
Subject(s): Genre/Form: Additional physical formats: Print version:: Analysis in Integer and Fractional DimensionsDDC classification:
  • 515.2433
LOC classification:
  • QA403 .B54 2001
Online resources:
Contents:
Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- What the book is about -- I A Prologue: Mostly Historical -- II Three Classical Inequalities -- III A Fourth Inequality -- IV Elementary Properties of the Fréchet Variation - an Introduction to Tensor Products -- V The Grothendieck Factorization Theorem -- VI An Introduction to Multidimensional Measure Theory -- VII An Introduction to Harmonic Analysis -- VIII Multilinear Extensions of the Grothendieck Inequality (via… -- IX Product Fréchet measures -- X Brownian Motion and the Wiener Process -- XI Integrators -- XII A '3/2-dimensional' Cartesian Product -- XIII Fractional Cartesian Products and Combinatorial Dimension -- XIV The Last Chapter: Leads and Loose Ends -- Conventions and Notations -- Acknowledgements -- I A Prologue: Mostly Historical -- 1 From the Linear to the Bilinear -- 2 A Bilinear Theory -- 3 More of the Bilinear -- 4 From Bilinear to Multilinear and Fraction-linear -- Exercises -- Hints for Exercises in Chapter I -- II Three Classical Inequalities -- 1 Mise en Scéne: Rademacher Functions -- 2 The Khintchin… -- 3 The Littlewood and Orlicz Mixed-norm Inequalities -- 4 The Three Inequalities are Equivalent -- 5 An Application: Littlewood's 4/3-inequality -- 6 General Systems and Best Constants -- Exercises -- Hints for Exercises in Chapter II -- III A Fourth Inequality -- 1 Mise en Scéne: Does the Khintchin… -- 2 An Elementary Proof -- 3 A Second Elementary Proof -- 4… -- Remarks: -- 5 A Representation of an Inner Product in a Hilbert Space -- 6 Comments (Mainly Historical) and Loose Ends -- Is the Grothendieck Inequality Equivalent to… -- More about the Inequality -- Exercises -- Hints for Exercises in Chapter III -- IV Elementary Properties of the Fréchet Variation - an Introduction to Tensor Products -- 1 Mise en Scéne: The Space….
2 Examples -- 3 Finitely Supported Functions are Norm-dense in… -- 4 Two Consequences -- A Fubini-type Property -- Remarks: -- 5 The Space… -- Remarks -- 6 A Brief Introduction to General Topological Tensor Products -- 7 A Brief Introduction to Projective Tensor Algebras -- 8 A Historical Backdrop -- A Brief Critique and a Preview -- Exercises -- Hints for Exercises in Chapter IV -- V The Grothendieck Factorization Theorem -- 1 Mise en Scéne: Factorization in One Dimension -- 2 An Extension to Two Dimensions -- 3 An Application -- 4 The g-norm -- 5 The g-norm in the Multilinear Case -- Exercises -- Hints for Exercises in Chapter V -- VI An Introduction to Multidimensional Measure Theory -- 1 Mise en Scéne: Fréchet Measures -- 2 Examples -- 3 The Fréchet Variation -- 4 An Extension Theorem -- 5 Integrals with Respect to F-measures -- 6 The Projective Tensor Algebra… -- 7 A Multilinear Riesz Representation Theorem -- 8 A Historical Backdrop -- Exercises -- Hints for Exercises in Chapter VI -- VII An Introduction to Harmonic Analysis -- 1 Mise en Scéne: Mainly a Historical Perspective -- 2 The Setup -- A Compact Abelian Group and its Dual -- Convolution -- Haar Measure -- 3 Elementary Representation Theory -- Remarks: -- 4 Some History -- 5 Analysis of Walsh Systems: A First Step -- 6 W is a Rosenthal Set -- 7 Restriction Algebras -- 8 Harmonic Analysis and Tensor Analysis -- Remarks: -- 9 Bonami's Inequalities: A Measurement of Complexity -- Remarks: -- 10 The Littlewood 2n/(n + 1)-Inequalities: Another Measurement of Complexity -- Remarks: -- 11 p-Sidon Sets -- Remarks: -- Remarks: -- 12 Transcriptions -- Dissociate Sets - Definition and Examples -- Riesz Products -- Restriction Algebras and Tensor Algebras -- The Rosenthal Property -- Bonami's Inequalities -- p-Sidonicity -- Exercises -- Hints for Exercises in Chapter VII.
VIII Multilinear Extensions of the Grothendieck Inequality (via… -- 1 Mise en Scène: A Basic Issue -- The Factorization Theorem (cf. Theorem IV.2) -- 2 Projective Boundedness -- 3 Uniformizable… -- Remarks: -- 4 A Projectively Bounded Trilinear Functional -- 5 A Characterization -- 6 Projectively Unbounded Trilinear Functionals -- Remarks: -- 7 The General Case -- 8… -- 9 Proof of Theorem 19 -- Remarks: -- Exercises -- Hints for Exercises in Chapter VIII -- IX Product Fréchet Measures -- 1 Mise en Scène: A Basic Question -- 2 A Preview -- 3 Projective Boundedness -- 4 Every µ 2 F is Projectively Bounded -- 5 There Exist Projectively Unbounded F-measures -- 6 Projective Boundedness in Topological Settings -- Remarks: -- 7 Projective Boundedness in Topological-group Settings -- Remarks: -- 8 Examples -- Remarks: -- Exercises -- Hints for Exercises in Chapter IX -- X Brownian Motion and the Wiener Process -- 1 Mise en Scène: A Historical Backdrop and Heuristics -- From Brown to Wiener -- Heuristics -- Remarks: -- 2 A Mathematical Model for Brownian Motion -- A Construction of a Wiener Process -- 3 The Wiener Integral -- Remarks: -- Remarks: -- 4 Sub-Gaussian Systems -- Remarks: -- 5 Random Series -- Remarks: -- 6 Variations of the Wiener F-measures -- 7 A Multiple Wiener Integral -- 8 The Beginning of Adaptive Stochastic Integration -- Remarks: -- 9 Sub-Alpha-systems -- Remarks: -- 10 Measurements of Stochastic Complexity -- 11 The nth Wiener Chaos Process and its Associated F-measure -- 12 Mise en Scène (1 continued): Further Approximations of Brownian Motion -- 13 Random Walks and Decision Making Machines -- Remarks: -- 14 Apha-Chaos: A Definition, a Limit Theorem, and Some Examples -- Examples -- Questions -- Exercises -- Hints for Exercises in Chapter X -- XI Integrators -- 1 Mise en Scène: A General View -- 2 Integrators and Integrals -- Remarks:.
3 Examples -- Remarks: -- Remark -- Remarks: -- Remarks: -- 5 Two Questions - a Preview -- Remarks: -- 6 An Application of the Grothendieck Factorization Theorem -- Remarks: -- 7 Integrators Indexed by n-dimensional Sets -- Remarks: -- 8 Examples: Random Constructions -- 9 Independent Products of Integrators -- 10 Products of a Wiener Process -- 11 Random Integrands in One Parameter -- Via Riemann Sums -- Via Stochastic Series -- Remarks: -- Exercises -- Hints for Exercises in Chapter XI -- XII A '3/2-dimensional' Cartesian Product -- 1 Mise en Scène: Two Basic Questions -- 2 A Littlewood Inequality in 'Dimension' 3/2 -- 3 A Khintchin Inequality in 'Dimension' 3/2 -- Remark: -- Remarks: -- 4 Tensor Products in 'Dimension' 3/2 -- Remarks: -- 5 Fréchet Measures in 'Dimension' 3/2 -- 6 Product F-measures and Projective Boundedness in 'Dimension' 3/2 -- Exercises -- Hints for Exercises in Chapter XII -- XIII Fractional Cartesian Products and Combinatorial Dimension -- 1 Mise en Scène: Fractional Products -- 2 A Littlewood Inequality in Fractional 'Dimension' -- Remarks: -- 3 A Khintchin Inequality in Fractional 'Dimension' -- Remarks: -- 4 Combinatorial Dimension -- Remarks: -- 5 Fractional Cartesian Products are q-products -- Remarks: -- 6 Random Constructions -- Remarks: -- 7 A Relation between the dim-scale and the Sigma-scale -- Remarks: -- Remarks: -- Remarks: -- 8 A Relation between the dim-scale and the Delta-scale -- Remarks: -- Exercises -- Hints for Exercises in Chapter XIII -- XIV The Last Chapter: Leads and Loose Ends -- 1 Mise en Scène: The Last Chapter -- 2 Fréchet Measures in Fractional Dimensions -- Remarks: -- Remarks: -- 3 Combinatorial Dimension in Topological and Measurable Settings -- Remarks: -- 4 Harmonic Analysis -- Convolution -- Examples -- 5 Random Walks -- Remarks: -- 6 Alpha-chaos -- Existence -- Detection -- Remarks:.
7 Integrators in Fractional Dimensions -- Exercises -- Hints for Exercises in Chapter XIV -- References -- Index.
Summary: Thorough and self-contained study for graduate students and researchers.
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Cover -- Half-title -- Series-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- What the book is about -- I A Prologue: Mostly Historical -- II Three Classical Inequalities -- III A Fourth Inequality -- IV Elementary Properties of the Fréchet Variation - an Introduction to Tensor Products -- V The Grothendieck Factorization Theorem -- VI An Introduction to Multidimensional Measure Theory -- VII An Introduction to Harmonic Analysis -- VIII Multilinear Extensions of the Grothendieck Inequality (via… -- IX Product Fréchet measures -- X Brownian Motion and the Wiener Process -- XI Integrators -- XII A '3/2-dimensional' Cartesian Product -- XIII Fractional Cartesian Products and Combinatorial Dimension -- XIV The Last Chapter: Leads and Loose Ends -- Conventions and Notations -- Acknowledgements -- I A Prologue: Mostly Historical -- 1 From the Linear to the Bilinear -- 2 A Bilinear Theory -- 3 More of the Bilinear -- 4 From Bilinear to Multilinear and Fraction-linear -- Exercises -- Hints for Exercises in Chapter I -- II Three Classical Inequalities -- 1 Mise en Scéne: Rademacher Functions -- 2 The Khintchin… -- 3 The Littlewood and Orlicz Mixed-norm Inequalities -- 4 The Three Inequalities are Equivalent -- 5 An Application: Littlewood's 4/3-inequality -- 6 General Systems and Best Constants -- Exercises -- Hints for Exercises in Chapter II -- III A Fourth Inequality -- 1 Mise en Scéne: Does the Khintchin… -- 2 An Elementary Proof -- 3 A Second Elementary Proof -- 4… -- Remarks: -- 5 A Representation of an Inner Product in a Hilbert Space -- 6 Comments (Mainly Historical) and Loose Ends -- Is the Grothendieck Inequality Equivalent to… -- More about the Inequality -- Exercises -- Hints for Exercises in Chapter III -- IV Elementary Properties of the Fréchet Variation - an Introduction to Tensor Products -- 1 Mise en Scéne: The Space….

2 Examples -- 3 Finitely Supported Functions are Norm-dense in… -- 4 Two Consequences -- A Fubini-type Property -- Remarks: -- 5 The Space… -- Remarks -- 6 A Brief Introduction to General Topological Tensor Products -- 7 A Brief Introduction to Projective Tensor Algebras -- 8 A Historical Backdrop -- A Brief Critique and a Preview -- Exercises -- Hints for Exercises in Chapter IV -- V The Grothendieck Factorization Theorem -- 1 Mise en Scéne: Factorization in One Dimension -- 2 An Extension to Two Dimensions -- 3 An Application -- 4 The g-norm -- 5 The g-norm in the Multilinear Case -- Exercises -- Hints for Exercises in Chapter V -- VI An Introduction to Multidimensional Measure Theory -- 1 Mise en Scéne: Fréchet Measures -- 2 Examples -- 3 The Fréchet Variation -- 4 An Extension Theorem -- 5 Integrals with Respect to F-measures -- 6 The Projective Tensor Algebra… -- 7 A Multilinear Riesz Representation Theorem -- 8 A Historical Backdrop -- Exercises -- Hints for Exercises in Chapter VI -- VII An Introduction to Harmonic Analysis -- 1 Mise en Scéne: Mainly a Historical Perspective -- 2 The Setup -- A Compact Abelian Group and its Dual -- Convolution -- Haar Measure -- 3 Elementary Representation Theory -- Remarks: -- 4 Some History -- 5 Analysis of Walsh Systems: A First Step -- 6 W is a Rosenthal Set -- 7 Restriction Algebras -- 8 Harmonic Analysis and Tensor Analysis -- Remarks: -- 9 Bonami's Inequalities: A Measurement of Complexity -- Remarks: -- 10 The Littlewood 2n/(n + 1)-Inequalities: Another Measurement of Complexity -- Remarks: -- 11 p-Sidon Sets -- Remarks: -- Remarks: -- 12 Transcriptions -- Dissociate Sets - Definition and Examples -- Riesz Products -- Restriction Algebras and Tensor Algebras -- The Rosenthal Property -- Bonami's Inequalities -- p-Sidonicity -- Exercises -- Hints for Exercises in Chapter VII.

VIII Multilinear Extensions of the Grothendieck Inequality (via… -- 1 Mise en Scène: A Basic Issue -- The Factorization Theorem (cf. Theorem IV.2) -- 2 Projective Boundedness -- 3 Uniformizable… -- Remarks: -- 4 A Projectively Bounded Trilinear Functional -- 5 A Characterization -- 6 Projectively Unbounded Trilinear Functionals -- Remarks: -- 7 The General Case -- 8… -- 9 Proof of Theorem 19 -- Remarks: -- Exercises -- Hints for Exercises in Chapter VIII -- IX Product Fréchet Measures -- 1 Mise en Scène: A Basic Question -- 2 A Preview -- 3 Projective Boundedness -- 4 Every µ 2 F is Projectively Bounded -- 5 There Exist Projectively Unbounded F-measures -- 6 Projective Boundedness in Topological Settings -- Remarks: -- 7 Projective Boundedness in Topological-group Settings -- Remarks: -- 8 Examples -- Remarks: -- Exercises -- Hints for Exercises in Chapter IX -- X Brownian Motion and the Wiener Process -- 1 Mise en Scène: A Historical Backdrop and Heuristics -- From Brown to Wiener -- Heuristics -- Remarks: -- 2 A Mathematical Model for Brownian Motion -- A Construction of a Wiener Process -- 3 The Wiener Integral -- Remarks: -- Remarks: -- 4 Sub-Gaussian Systems -- Remarks: -- 5 Random Series -- Remarks: -- 6 Variations of the Wiener F-measures -- 7 A Multiple Wiener Integral -- 8 The Beginning of Adaptive Stochastic Integration -- Remarks: -- 9 Sub-Alpha-systems -- Remarks: -- 10 Measurements of Stochastic Complexity -- 11 The nth Wiener Chaos Process and its Associated F-measure -- 12 Mise en Scène (1 continued): Further Approximations of Brownian Motion -- 13 Random Walks and Decision Making Machines -- Remarks: -- 14 Apha-Chaos: A Definition, a Limit Theorem, and Some Examples -- Examples -- Questions -- Exercises -- Hints for Exercises in Chapter X -- XI Integrators -- 1 Mise en Scène: A General View -- 2 Integrators and Integrals -- Remarks:.

3 Examples -- Remarks: -- Remark -- Remarks: -- Remarks: -- 5 Two Questions - a Preview -- Remarks: -- 6 An Application of the Grothendieck Factorization Theorem -- Remarks: -- 7 Integrators Indexed by n-dimensional Sets -- Remarks: -- 8 Examples: Random Constructions -- 9 Independent Products of Integrators -- 10 Products of a Wiener Process -- 11 Random Integrands in One Parameter -- Via Riemann Sums -- Via Stochastic Series -- Remarks: -- Exercises -- Hints for Exercises in Chapter XI -- XII A '3/2-dimensional' Cartesian Product -- 1 Mise en Scène: Two Basic Questions -- 2 A Littlewood Inequality in 'Dimension' 3/2 -- 3 A Khintchin Inequality in 'Dimension' 3/2 -- Remark: -- Remarks: -- 4 Tensor Products in 'Dimension' 3/2 -- Remarks: -- 5 Fréchet Measures in 'Dimension' 3/2 -- 6 Product F-measures and Projective Boundedness in 'Dimension' 3/2 -- Exercises -- Hints for Exercises in Chapter XII -- XIII Fractional Cartesian Products and Combinatorial Dimension -- 1 Mise en Scène: Fractional Products -- 2 A Littlewood Inequality in Fractional 'Dimension' -- Remarks: -- 3 A Khintchin Inequality in Fractional 'Dimension' -- Remarks: -- 4 Combinatorial Dimension -- Remarks: -- 5 Fractional Cartesian Products are q-products -- Remarks: -- 6 Random Constructions -- Remarks: -- 7 A Relation between the dim-scale and the Sigma-scale -- Remarks: -- Remarks: -- Remarks: -- 8 A Relation between the dim-scale and the Delta-scale -- Remarks: -- Exercises -- Hints for Exercises in Chapter XIII -- XIV The Last Chapter: Leads and Loose Ends -- 1 Mise en Scène: The Last Chapter -- 2 Fréchet Measures in Fractional Dimensions -- Remarks: -- Remarks: -- 3 Combinatorial Dimension in Topological and Measurable Settings -- Remarks: -- 4 Harmonic Analysis -- Convolution -- Examples -- 5 Random Walks -- Remarks: -- 6 Alpha-chaos -- Existence -- Detection -- Remarks:.

7 Integrators in Fractional Dimensions -- Exercises -- Hints for Exercises in Chapter XIV -- References -- Index.

Thorough and self-contained study for graduate students and researchers.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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