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Blei R. — Analysis in Integer and Fractional Dimensions
Blei R. — Analysis in Integer and Fractional Dimensions



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Название: Analysis in Integer and Fractional Dimensions

Автор: Blei R.

Аннотация:

This book provides a thorough and self-contained study of interdependence and complexity in settings of functional analysis, harmonic analysis and stochastic analysis. It focuses on "dimension" as a basic counter of degrees of freedom, leading to precise relations between combinatorial measurements and various indices originating from the classical inequalities of Khintchin, Littlewood and Grothendieck. Topics include the (two-dimensional) Grothendieck inequality and its extensions to higher dimensions, stochastic models of Brownian motion, degrees of randomness and Fréchet measures in stochastic analysis. This book is primarily aimed at graduate students specializing in harmonic analysis, functional analysis or probability theory. It contains many exercises and is suitable as a textbook. It is also of interest to computer scientists, physicists, statisticians, biologists and economists.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2001

Количество страниц: 576

Добавлена в каталог: 22.05.2005

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Предметный указатель
Independent products of integrators      Chapter XI $\S6$
Injective tensor norm      60 82 see
Inner product in Grothendieck's inequality      9 13 17 see
Integral with respect to integrator      see also Adaptive stochastic integration
Integral with respect to integrator, $I_X(.)$ (functional-analytic approach)      351
Integral with respect to integrator, $\int_{[0,1]}dX$ (measure-theoretic approach)      353
Integral with respect to integrator, multi-parameter case (preview)      Chapter XI $\S5$
Integration by parts formula      411
Integration with respect to 'n—dimensional' 1-process      396
Integration with respect to Frechet measures      Chapter VI $\S5$ 122 130—132 506—507
Integrator      Chapter XI $\S1$ Chapter
Interchange of limit and integration      417
Interdependence      see also Complexity
Interdependence as complexity      187—188
Interdependence in random walks      Chapter X $\S13$
Interdependence of elementary tensors      186—187
Interdependence, conveyed by tail-probabilities      297—299 Chapter 364
Interdependence, functional      175
Interdependence, marked by 'dimension'      Chapter XII $\S1$ Chapter
Interdependence, marked by 'type'      444—447
Interdependence, marked by combinatorial dimension      477—478 517
Interdependence, marked by dimension of 1-process      503 530—531
Interdependence, measurements      187
Interdependence, statistical      175
Iterated integral Ito integral      316—317
Iterated integral with respect to Frechet measure      120
Iterated integral with respect to n-process      379
Iterated integral with respect to product Frechet measures      263—264
Iterated integral, Lebesgue — Stieltjes      131—132 (Exercises 15 16)
Iterated integral, Riemann—Stieltjes      3
Ito integral: as iterated integral      Chapter X $\S8$
Ito integral: via measure-theoretic approach      405—408
Ito's formula      318 343 409
k-disjoint rectangles      63
Kahane — Salem — Zygmund estimates      301
Khintchin $L^1$-$L^2$ inequality      see also $\Lambda(2)$-set problem Chapter
Khintchin $L^1$-$L^2$ inequality for general systems      30
Khintchin $L^1$-$L^2$ inequality in dimension n      202 (Exercise 32)
Khintchin $L^1$-$L^2$ inequality, equivalence to Littlewood's and Orlicz's mixed-norm inequalities      Chapter II $\S4$
Khintchin $L^1$-$L^2$ inequality, history      22—23
Khintchin $L^1$-$L^2$ inequality, statement, proof, and history      Chapter II $\S2$
Khintchin $L^1$-$L^2$ inequality, upgraded      40 46 see
Khintchin inequalities in dimension 3/2      Chapter XII $\S3$
Khintchin inequalities in fractional dimensions      Chapter XIII $\S3$
Khintchin inequalities, application to $\Lambda(2)$-uniformizing constants      49
Khintchin inequalities, history and impact      22 55 171—174
Khintchin inequalities, proofs      32 (Exercise 3) 342
Kronecker product      249
Lacunary      12 53 150 173 185 189 191 198 199 424
Law of the iterated logarithm      22 171—172
Limit theorems      335 338 521—522
Linear programming problem (associated with a fractional Cartesian product)      478
Littlewood 2n/(n+1)-inequality, an extension of Littlewood's 4/3-inequality      10—11
Littlewood 2n/(n+1)-inequality, application in harmonic analysis      185
Littlewood 2n/(n+1)-inequality, application in tensor analysis      185—186
Littlewood 2n/(n+1)-inequality, calibrating Plancherel's theorem      187
Littlewood 2n/(n+1)-inequality, marking functional interdependence      175 188—189
Littlewood 2n/(n+1)-inequality, statement and proof      Chapter VII $\S10$
Littlewood index I (of): $\alpha$-chaos      364
Littlewood index I (of): $\Lambda(q)$-process      369
Littlewood index I (of): $\Lambda(q)^{#}$-process      372
Littlewood index I (of): 1-integrator      356
Littlewood index I (of): F-measure      507
Littlewood index I (of): nth Wiener chaos process      325
Littlewood index I (of): p-stable motion      374
Littlewood index I (of): U-integrator      392 528—529
Littlewood index I (of): Wiener process      307 310
Littlewood inequality in dimension 3/2 application to tensor products      443
Littlewood inequality in dimension 3/2 application to tensor products, preview      185—187
Littlewood inequality in dimension 3/2 application to tensor products, statement and proof      Chapter XII $\S2$
Littlewood inequality in fractional key in observing 'd-dimensional' 1—processes      531
Littlewood inequality in fractional key in observing 'd-dimensional' 1—processes, statement and proof      Chapter XIII $\S2$
Littlewood mixed-norm inequality      see also Orlicz's mixed-norm inequality
Littlewood mixed-norm inequality in Orlicz's paper      33 (Exercise 5)
Littlewood mixed-norm inequality in stochastic setting      371
Littlewood mixed-norm inequality, equivalence to Khintchin $L^1$-$L^2$ and Orlicz's mixed-norm inequalities      Chapter II $\S4$
Littlewood mixed-norm inequality, extensions      176 463—464 489
Littlewood mixed-norm inequality, feasibility of extension      17 (Exercise 6)
Littlewood mixed-norm inequality, precursor to and instance of Grothendieck's inequality      7 11 17 39—40
Littlewood mixed-norm inequality, reformulation      92 (Exercise 19 i)
Littlewood mixed-norm inequality, statement and proof      23—24
Littlewood mixed-norm inequality, use by Davie      11
Littlewood's 4/3-inequality, answer to question by Daniell      6—7
Littlewood's 4/3-inequality, application in harmonic analysis      178 185
Littlewood's 4/3-inequality, application in tensor analysis      62
Littlewood's 4/3-inequality, statement and proof      Chapter II $\S5$
Lower combinatorial dimension      476
m-linear Hoelder inequality      458—459
Martingales      146 196—197 see
Maximal cover (definition)      458
Maximal fractional Cartesian product (definition)      493 see
Mazur — Orlicz identity      163 383 see
Measure-theoretic approach (to stochastic integration) deterministic integrands      351
Measure-theoretic approach (to stochastic integration), Ito integral      405
Measure-theoretic approach (to stochastic integration), multi-parameter case      379 380
Measure-theoretic approach (to stochastic integration), p-stable motion      373
Measure-theoretic approach (to stochastic integration), via stochastic series      412
Minimal fractional Cartesian product (definition)      493
Mixed-norm inequalities      see Littlewood's mixed norm inequalities; Orlicz's mixed-norm inequalities
Mixed-norm space (in Grothendieck inequality)      51 251
Multi-measures      83 see
Multidimensional integral      379
Multilinear Grothendieck inequalities      Chapter VIII see
Multilinear Riesz representation theorem      Chapter VI $\S7$ see
Multiple Wiener integral      Chapter X $\S7$ see
Multiple Wiener — Ito integral      311
n-disjoint      485
n-process      343 (Exercise 28)
Nikodym boundedness principle      112 129
Non-adapted stochastic integrals      410
Non-anticipative stochastic normalization      331
Nowhere differentiability of sample paths of Wiener process      289
nth Wiener Chaos process $W_n$      Chapter X $\S11$
nth Wiener Chaos process $W_n$, associated $F_2$-measure $\mu\mathrm{w}_n$      325
nth Wiener Chaos process $W_n$, definition      325
nth Wiener Chaos process $W_n$, stochastic complexity of $W_n$      328
nth-Wiener chaos      Chapter X $\S11$ 338
Ogawa integral      418
Optimal $\mathfrak{D}$-type      see Type
Optimal $\mathfrak{F}$-type      see Type
Orlicz functions      297 307—308 321
Orlicz norms (marking stochastic complexity)      311 328—329
Orlicz's mixed-norm inequality, equivalence to Khintchin $L^1$-$L^2$ and Littlewood's mixed-norm inequalities      Chapter II $\S4$
Orlicz's mixed-norm inequality, in stochastic setting      369—370
Orlicz's mixed-norm inequality, multilinear extension      176
Orlicz's mixed-norm inequality, reformulation      92 (Exercise 19 iv)
Orlicz's mixed-norm inequality, statement, proof, and history      24—25 33
p-Sidon set      see also Sidon set
p-Sidon set in $\Gamma$      466—467
p-Sidon set in W      465—466
p-Sidon set, basic characterizations      182
p-Sidon set, combinatorial characterization problem      190
p-Sidon set, definition (exact, asymptotic)      182
p-Sidon set, existence problem (p-Sidon set problem)      13 189 428
p-Sidon set, finite union problem      190
p-Sidon set, historical comments      185
p-Sidon set, terminology      182
p-stable motion, definition      373
p-stable motion, integrator      373
p-stable motion, physical meaning      378—379
p-stable motion, properties      374
p-stable motion, variations of associated $F_2$-measure      374—378
p-variation      see also Littlewood index
p-variation of Frechet measures      128 507
p-variation of scalar functions on [0,1]      515
Paley ordering of Walsh system      146 196—197
Parseval's formula      16 (Exercise 4) 142
Pietsch factorization theorem      98 101 105
Pisier's theorem      173 190
Plancherel's theorem in $L^2$($\mathbb{Z}_N$, uniform measure)      16 (Exercise 4)
Plancherel's theorem in $L^2(\mathrm{T},\mathfrak{m})$      196 (Exercise 1)
Plancherel's theorem in $L^2(\Omega,\mathbb{P})$      144—145
Plancherel's theorem, calibration      187
Polarization identities      163 167 200 421 see
Product Frechet measure in dimension 3/2      Chapter XII $\S6$
Product Frechet measure in fractional dimensions      509—510
Product Frechet measure in stochastic setting      399
Product Frechet measure, definition      248
Product Frechet measure, link with Grothendieck-type inequalities      254
Products of $L^1$-bounded additive processes      408—409
Products of Wiener processes      402—408
Projective tensor algebra      Chapter IV $\S7$ Chapter 130 186
Projective tensor norm      61 81 82 87 121 260—261 see
Projective tensor product      see projective tensor algedra
Projectively unbounded F-measures      Chapter IX $\S5$ see
Projectively unbounded functionals      Chapter VIII $\S6$ see
Protectively bounded forms      see also Grothendieck-type inequalities
Protectively bounded forms, characterizations      208—209 Chapter 230
Protectively bounded forms, definition      13—14 208
Protectively bounded forms, general characterization (problem)      240—242
Protectively bounded Frechet measures      see also Grothendieck-type inequalities; product Frechet-measure; Protectively bounded forms
Protectively bounded Frechet measures in dimension 3/2      Chapter XII $\S6$
Protectively bounded Frechet measures in fractional dimensions      509—512
Protectively bounded Frechet measures in topological setting      Chapter IX $\S6$
Protectively bounded Frechet measures in topological-group setting      Chapter IX $\S7$
Protectively bounded Frechet measures, characterization      254
Protectively bounded Frechet measures, definition      253
Quadratic variation      290 317 340
Rademacher characters      146 see
Rademacher functions      see also Rademacher system
Rademacher functions, characters on $\Omega$      137
Rademacher functions, definition      2 19 146
Rademacher functions, statistically independent random variables      20—21
Rademacher series      53
Rademacher system      see also Rademacher functions
Rademacher system, definition      2 19
Rademacher system, generalizations      Chapter II $\S6$ 191
Rademacher system, independent system      20—21 139
Rademacher system, sub-Gaussian system      299
Random integrands      Chapter XI $\S11$
Random integrator      356
Random series      200—201 (Exercise 30) Chapter
Random walks      see also Simple random walks; Decision making machines
Random walks by drunks      Chapter X $\S13$
Random walks, F-walks      333—335 336 Chapter
Random walks, simplest model of Brownian motion      283—284 331
Randomness      281 311 348 355—357 361—362
Reduced fractional Cartesian products      493
Restriction algebras $A(F)^*=L_F^{\infty}$      158
Restriction algebras $C^*_F=B(F)^F$      158
Restriction algebras $L_F^1=Q^{\infty}(F)$      245 (Hint 4)
Restriction algebras (duals and preduals)      see also tensor representations of restriction algebras
Restriction algebras A(F)      158
Restriction algebras B(F)      157
Restriction algebras in harmonic analysis      Chapter VII $\S7$
Restriction algebras in harmonic and tensor analysis      Chapter VII $\S8$
Restriction algebras V_n|_F      163
Riemann — Lebesgue lemma      145
Riesz product $L^{\infty}$-version      53—54 173 203
Riesz product, construction      198—199 (Exercises 17 18)
Riesz product, expansion      144 192—193
Riesz product, first appearance      53
Riesz product, summability kernel      143—144
Riesz representation theorem in dimension 3/2      451 454
Riesz representation theorem in fractional dimensions      507
Riesz representation theorem, measure-theoretic version      1
Riesz representation theorem, multilinear form      126
Riesz representation theorem, original form      1
Riesz representation theorem, primal form      1—2 72
Riesz set      200 (Exercise 26)
Riesz's (M.) theorem      147 197
Rosenthal property $W_k$      Chapter VII $\S6$
Rosenthal property in general setting      194
Rosenthal property, equivalent to separability      166 199
Rosenthal property, products of Sidon sets      72
Rosenthal set      151 see
Sample-path continuity of $\alpha$-chaos      366
Sample-path continuity of $\Lambda(5)$-process      368
Sample-path continuity of Wiener process      305
Sample-path continuity via stochastic series, entropy, majorizing measures and Kolmogorov's theorem      368
Scales (relations between) $\delta$-scale and $\xi$-scale (problem)      500
Scales (relations between) $\sigma$-scale and $\delta$-scale (problem)      500
Scales (relations between) dim-scale and $\delta$-scale      Chapter XIII $\S8$
Scales (relations between) dim-scale and $\sigma$-scale      Chapter XIII $\S7$
Schur product      272 (Exercise 2)
Schur property of $F_k$      69—72
Schur property of Sidon space      89 (Exercise 8)
Schur property, definition      69
Schur's theorem      69
Semi-martingale      409
Series, Fourier — Stieltjes series      136
Series, Fourier — Wiener series      294
Series, Haar — Wiener series      341—342 (Exercise 13)
Series, stochastic series approach to integration      412—419
Series, stochastic series approach to sample-path continuity      368
Series, stochastic series of a Wiener process      294—295 341
Series, stochastic series of integrator      387
Series, W-series      140
Series, Walsh series      147
Series, Walsh — Wiener series      341—342 (Exercise 13)
Sidon      see also p-Sidon set; Pisier's theorem
Sidon exponent      12 182 428
Sidon sequence      89 (Exercise 8)
Sidon set (definition)      12 145 150
Sidon set is sub-Gaussian      300
Sidon space      89 (Exercise 8)
Sidon, combinatorial characterization of Sidon sets      190
Sidon, finite union of Sidon sets      190
Sidon, Sidon's theorem      12 150
Sidon, Sidonicity and functional independence      188—189
Simple random walk      see also Random walks
Simple random walk by drunk      331
Simple random walk, approximation to Brownian movement      283—284 329
Simple random walk, instance of F-walks      333
Skorohod integral      417—418
Spectrum      145
Standard $\alpha$-variable      335 524 see
Standard step function      312
Standard Sub-$\alpha$-variable      522 see
Steinhaus functions in Littlewood's work      22 172
Steinhaus functions, definition      30
Steinhaus functions, dubbed by Salem and Zygmund      30 172 300
Steinhaus functions, independent characters      192 436
Steinhaus functions, involving inequalities equivalent to Khintchin's      172 195
Steinhaus system      see Steinhaus functions
Stochastic complexity in $W_n$      328—329
Stochastic complexity measurements      Chapter X $\S10$
Stochastic complexity of $\alpha$-chaos      348 364
Stochastic complexity of Brownian displacements      348
Stochastic complexity of random walks      333—334 522
Stochastic complexity, conveyed by variations of F-measures      355—357
Stochastic complexity, detecting complexity of Brownian movement      338—339
Stochastic complexity, Wiener process is least complex      311 328 357
Stone Cech compactification      93 (Hint 12)
Stratonovich integral      407 416
Strongly disjoint sets      153
Sub-$\alpha$-system      321 see
Sub-$\alpha$-variable      321 see
Sub-Gaussian systems      Chapter X $\S4$
Summability kernel      142
Sup-norm partition      467
Superior integral      128—129 133
Symmetric functions that vanish on the 'hyper-diagonals'      313
Symmetric functions that vanish on the diagonal      312
Symmetric n-arrays      162
Symmetrization in $[0,1]^2$      312
Symmetrization in $[0,1]^n$      314
Symmetrization, role in stochastic integration      315—316
Tail-probabilities      see also Complexity; Stochastic complexity; Interdependence
Tail-probabilities, exponential estimates      363—364 524
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