Analysis in Integer and Fractional Dimensions
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1 Analysis in Integer and Fractional Dimensions Ron Blei University of Connecticut ^ i UNIVERSITY PRE^S
2 Contents Preface Acknowledgements I A Prologue: Mostly Historical 1 1. From the Linear to the Bilinear 1 2. A Bilinear Theory 6 3. More of the Bilinear 8 4. Prom Bilinear to Multilinear and Fraction-linear 10 Exercises 15 Hints for Exercises in Chapter I 18 II Three Classical Inequalities Mise en Scene: Rademacher Functions The Khintchin L 1 -!, 2 Inequality The Littlewood and Orlicz Mixed-norm Inequalities The Three Inequalities are Equivalent An Application: Littlewood's 4/3-inequality General Systems and Best Constants 28 "Exercises 31 Hints for Exercises in Chapter II 36 III A Fourth Inequality Mise en Scene: Does the Khintchin L 1 -!, 2 Inequality Imply the Grothendieck Inequality? An Elementary Proof A Second Elementary Proof A(2)-uniformizability A Representation of an Inner Product in a Hilbert Space Comments (Mainly Historical) and Loose Ends 53 xiii xix vn
3 viii Contents Exercises 57 Hints for Exercises in Chapter III 58 IV Elementary Properties of the Frechet ~ Variation - an Introduction to Tensor Products Mise en Scene: The Space F k (N,...,N) Examples Finitely Supported Functions are Norm-dense in F fe (N,...,N) Two Consequences The Space V k (N,...,N) A Brief Introduction to General Topological Tensor Products A Brief Introduction to Projective Tensor Algebras A Historical Backdrop 86 Exercises 88 Hints for Exercises in Chapter IV 92 V The Grothendieck Factorization Theorem Mise en Scene: Factorization in One Dimension > An Extension to Two Dimensions An Application The ff-norm The g-notm in the Multilinear Case 103 Exercises 105 Hints for Exercises in Chapter V 106 VI An Introduction to Multidimensional Measure Theory Mise en Scene: Frechet Measures Examples The Frechet Variation Ill 4. An Extension Theorem Integrals with Respect to F n -measures The Projective Tensor Algebra V^Ci,..., C n ) A Multilinear Riesz Representation Theorem A Historical Backdrop 126 Exercises 129 Hints for Exercises in Chapter VI 133
4 Contents ix VII An Introduction to Harmonic Analysis Mise en Scene: Mainly a Historical Perspective The Setup Elementary Representation Theory Some History Analysis of Walsh Systems: a First Step Wk is a Rosenthal Set Restriction Algebras Harmonic Analysis and Tensor Analysis Bonami's Inequalities: A Measurement of Complexity The Littlewood 2n/(n + 1)-Inequalities: Another Measurement of Complexity p-sidon Sets Transcriptions 190 Exercises 196 Hints for Exercises in Chapter VII 202 VIII Multilinear Extensions of the Grothendieck Inequality (via A(2)-uniformizability) Mise en Scene: A Basic Issue Projective Boundedness, Uniformizable A(2)-sets A Projectively Bounded Trilinear Functional A Characterization Projectively Unbounded Trilinear Functional The General Case if ~ Proof of Theorem Exercises 242 '"Hints for Exercises in Chapter VIII 245 IX Product Frechet Measures Mise en Scene: A Basic Question A Preview Projective Boundedness Every n F 2 is Projectively Bounded There Exist Projectively Unbounded ^-measures Projective Boundedness in Topological Settings Projective Boundedness in Topological-group Settings.v, 264
5 x Contents 8. Examples 269 Exercises 272 Hints for Exercises in Chapter IX : X Brownian Motion and the Wiener Process Mise en Scene: A Historical Backdrop and Heuristics A Mathematical Model for Brownian Motion The Wiener Integral Sub-Gaussian Systems Random Series Variations of the Wiener F 2 -measure A Multiple Wiener Integral The Beginning of Adaptive Stochastic Integration Sub-a-systems Measurements of Stochastic Complexity The nth Wiener Chaos Process and its Associated F-measure Mise en Scene ( 1 continued): Further Approximations of Brownian Motion Random Walks and Decision MakingNMachines a-chaos: A Definition, a Limit Theorem, and Some Examples 335 Exercises 339 Hints for Exercises in Chapter X 344 XI Integrators Mise en Scene: A General View Integrators and Integrals Examples More Examples: a-chaos, A(g)-processes, p-stable Motions Two Questions - a Preview An Application of the Grothendieck Factorization Theorem Integrators Indexed by n-dimensional Sets Examples: Random Constructions Independent Products of Integrators Products.of a Wiener Process Random Integrands in One Parameter 409
6 Contents Exercises 419 Hints for Exercises in Chapter XI 424 XII A '3/2-dimensional' Cartesian Product Mise en Scene: Two Basic Questions A Littlewood Inequality in 'Dimension' 3/ A Khintchin Inequality in 'Dimension' 3/ Tensor Products in 'Dimension' 3/ Frechet Measures in 'Dimension' 3/ Product -F-measures and Projective Boundedness in 'Dimension' 3/2 451 Exercises 453 Hints for Exercises in Chapter XII 455 XIII Fractional Cartesian Products and Combinatorial Dimension Mise en Scene: Fractional Products A Littlewood Inequality in Fractional 'Dimension' A Khintchin Inequality in Fractional 'Dimension' Combinatorial Dimension Fractional Cartesian Products are g-products Random Constructions A Relation between the dim-scale and the cr-scale A Relation between the dim-scale and the 5-scale 495 Exercises 500 Hints for Exercises in Chapter XIII 501 XIV The Last Chapter: Leads and Loose Ends Mise en Scene: The Last Chapter Frechet Measures in Fractional Dimensions Combinatorial Dimension in Topological ~" and Measurable Settings Harmonic Analysis Random Walks a-chaos Integrators in Fractional Dimensions 528 Exercises 531 Hints for Exercises in Chapter XIV 533 References 534 Index 547 xi
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