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



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Ќазвание: Analysis in Integer and Fractional Dimensions

јвтор: Blei R.

јннотаци€:

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

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2001

 оличество страниц: 576

ƒобавлена в каталог: 22.05.2005

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ѕредметный указатель
$A(\Omega)$, absolutely convergent W-series      150
$F_k$-measures      see Frechet measures
$F_k$-variation      see Frechet variation
$F_U$-measure associated with multi-parameter process      Chapter XI $\S7$
$F_{n\sigma}$ $\Lamda(q)$-process      369Ч370
$F_{n\sigma}$ (symmetric elements in $F_n$, that vanish on diagonals)      162Ч163
$F_{n\sigma}$ independent product of 1-integrators      Chapter XI $\S9$
$F_{n\sigma}$ integrator      351Ч353
$F_{n\sigma}$, $\alpha$-chaos      337Ч338
$F_{n\sigma}$, generalized Wiener process      293Ч294
$F_{n\sigma}$, p-stable motion      373Ч374
$F_{n\sigma}$, Wiener process      110Ч111 292Ч293
$H_n$      see also nth Wiener Chaos process
$H_n$, $\delta_{H_n}$      323
$H_n$, definition      320
$H_n$, properties      324
$I_W$ (definition)      290 see
$I_X$ (definition)      358
$I_{W_n}$ (definition)      Chapter X $\S7$ see
$L^1$-bounded additive process      363 381 382 385 387 408 422 30) 423
$L^2$-bounded process with orthogonal increments      357 359Ч360 379 388 419 422
$L^2$-factorizable      271 276
$L^p$-bounded martingales      363 387 (Exercise
$\alpha$-chaos, construction of exact n-chaos      338
$\alpha$-chaos, definition      337 364
$\alpha$-chaos, detection      526 528
$\alpha$-chaos, for non-integer $\alpha$ (question)      338
$\alpha$-chaos, general constructions      420 (Exercise 17) Chapter
$\alpha$-chaos, homogeneous integrator      364 337 356Ч357
$\alpha$-chaos, marking randomness      348
$\alpha$-chaos, variations of associated F-measures      337 356Ч357
$\alpha$-product, definition      475Ч476
$\alpha$-product, deterministic constructions      Chapter XIII $\S5$
$\alpha$-product, random constructions      Chapter XIII $\S6$
$\alpha$-system (exact, asymptotic)      see also $\alpha$-chaos; $\alpha$-variable; sub-$\alpha$-variable
$\alpha$-system (exact, asymptotic), examples      525Ч526
$\alpha$-variable (exact, asymptotic, standard)      see also Sub-$\alpha$-variable
$\alpha$-variable (exact, asymptotic, standard) in limit theorems      335 524
$\alpha$-variable (exact, asymptotic, standard), definition      335
$\alpha(U)$ (optimal value of linear programming problem)      478
$\delta$-scale 'continuously' calibrated?      175 434
$\delta$-scale applied to $E_n$      195Ч196
$\delta$-scale applied to $W_n$      174Ч175
$\delta$-scale applied to 3/2-fold Cartesian product      Chapter XII $\S3$
$\delta$-scale applied to general fractional Cartesian products      Chapter XIII $\S3$
$\delta$-scale applied to homogeneous integrator      364
$\delta$-scale in harmonic analysis setting      174 434
$\delta$-scale in probability theory setting      Chapter X $\S10$
$\delta$-scale measuring complexity      175
$\delta$-scale measuring interdependence      Chapter X $\S9$
$\delta$-scale, relation to $\sigma$-scale      
440 500
$\delta$-scale, relation to dim-scale      Chapter XIII $\S8$
$\eta$-measurement      see also $\delta$-scale
$\eta$-measurement in harmonic analysis setting      174 434
$\eta$-measurement in probability theory setting      298
$\Lambda(2)$-set problem      56
$\Lambda(2)$-set union problem      56Ч57
$\Lambda(2)$-space      46 see
$\Lambda(2)$-uniformizability      40 53 54 56Ч57 Chapter Chapter 214 241
$\Lambda(2)$-uniformizing constants      210
$\Lambda(2)$-uniformizing map      210
$\Lambda(p)$-set definition      55 173Ч174 202
$\Lambda(p)$-set problem      56 338 524
$\Lambda(p)$-space      55 58
$\Lambda(q)$-process, $\Lambda(q)^{#}$-process and the '$\Lambda(q)$-set problem'      372
$\Lambda(q)$-process, $\Lambda(q)^{#}$-process and the '$\Lambda(q)$-set problem' for $q\leq2$      372
$\Lambda(q)$-process, $\Lambda(q)^{#}$-process and the '$\Lambda(q)$-set problem', constructions      366 (Exercise
$\Lambda(q)$-process, $\Lambda(q)^{#}$-process and the '$\Lambda(q)$-set problem', definition      366
$\Lambda(q)$-system      366
$\lambda_X$      (measure associated with $L^2$-bounded process)
$\mathfrak(F)$-type      see Type
$\mu_X$ (F-measure associated with process X; definitions) 1-integrator X      351
$\mu_X$ (F-measure associated with process X; definitions) 1-process X      395 530
$\mu_X$ (F-measure associated with process X; definitions), U-integrator X      391
'Theoreme fondamental de la theorie metrique des produits tensoriels'      see Grothendieck's 'theoreme fondamental de la theorie metrique des produits tensoriels'
'Zero knowledge'      282 284 330
A(G), absolutely convergent $\hat G$-series      15
A(T), absolutely convergent Fourier series      149
Absolutely summing operators: definition      91Ч92
Absolutely summing operators: examples      92
Absolutely summing operators: study by Lindenstrauss and Pelczynski      54 87Ч88
Adaptive stochastic integration      290 317 Chapter 409Ч410
Algebraically independent spectral sets      see also Steinhaus system
Algebraically independent spectral sets, definition      191Ч192 436
Algebraically independent spectral sets, example      192 436
Ambient product      458 see
Atomic-molecular hypothesis      280 281
Banach's theorem      53 173
Bimeasures      see also Multi-measures
Bimeasures so dubbed      7
Bimeasures, historical connections      9 126Ч129
Bonami's inequalities, general setting      194
Bonami's inequalities, historical context      171Ч174
Bonami's inequalities, measurement of complexity      174Ч175
Bonami's inequalities, statement and proof      170Ч171
Bourgain's theorem      56 338 372 524
Brownian displacements in Einstein s model      Chapter X $\S12$
Brownian displacements in Wiener's model      Chapter X $\S12$
Brownian displacements, perceptions      282Ч283
Brownian motion      see also Wiener process
Brownian motion in the mathematical literature      306Ч307
Brownian motion, first and further approximations      Chapter X $\S12$
Brownian motion, heuristics leading to models      282Ч285 348
Brownian motion, history      279Ч282
Brownian motion, mathematical model (Wiener's)      Chapter X $\S2$
Brownian motion, stochastic complexity      338Ч339
Brownian movement      see Brownian motion
Brownian particle      282
Brownian sheet      293
Brownian trajectories as random walks      283Ч284 331 336
Brownian trajectories, assumptions about      282 329Ч330
Brownian trajectories, building blocks      xiii 73 135 187 191
Brownian trajectories, properties expressed by Wiener process      289Ч290
Burkholder Ч Gundy martingale inequalities      197 (Exercise 11) 299
Caratheodory Ч Hahn Ч Jordan theorem in multidimensional measure theory      Chapter VI $\S$4
Carleson's theorem      147
Ceneralized Minkowski inequality      25 32 459
Central limit theorem      284 322 329 330 335 336 343 345
Chaos      see $\alpha$-chaos; homogeneous chaos; Wiener Chaos
Characteristic function of normal r.v. (in proof of Grothendieck's inequality)      54
Characters of compact Abelian groups      137
Characters, $\mathbb{Z}_n$      16 (Exercise 4) 34
Characters, $\Omega$      138Ч139
Characters, $\Omega^n$      159
Circle group      29 55 109 136
Combinatorial complexity of random walks      333Ч334 521 524 528
Combinatorial device      488
Combinatorial dimension      see also fractional Cartesian products; $\alpha$-products
Combinatorial dimension in topological and measurable settings      Chapter XIV $\S3$
Combinatorial dimension, basic properties      476Ч477
Combinatorial dimension, definition (upper, lower, exact, asymptotic)      475Ч476
Combinatorial dimension, existence of $\alpha$-dimensional sets      477Ч478 Chapter
Combinatorial dimension, measurement of interdependence      456Ч458 477
Combinatorial dimension, motivation      186Ч188 433Ч434 436
Combinatorial measurements in Walsh system W      493Ч494
Combinatorial-harmonic analytic gauge $r_F$      435Ч436
Complete boundedness (relation to LЧfactorisability and convolvability)      271Ч272
complexity      see also interdependence; stochastic complexity; combinatorial complexity of random walks
Complexity 'hidden' in drunk's walks      334
Complexity, $\delta$-scale      174Ч175 Chapter
Complexity, $\sigma$-scale      188Ч189
Complexity, evolving in W      137 499Ч500
Continuous geometries      517Ч518
Convolution in $F(\Omega)$      520
Convolution in $F_2$ (preview)      78 Chapter
Convolution in $F_n$      Chapter IX $\S7$
Convolution in $L_1(\Omega,\mathbb{P})$      141
Convolution in $M(\Omega)$      139Ч140
Convolver definition      266
Convolver examples      Chapter IX $\S8$
Cover in definition of fractional Cartesian products      186Ч187 457Ч458
Cover in multilinear Grothendieck inequalities      227Ч228
Cover, k-cover      458
Cover, maximal k-cover      458
Cover, minimal k-cover      493
Crossnorms      81Ч82 see
Cylinder sets      202 (Hint 3) 518
Decision making machines      Chapter X $\S13$ 524 see
Decoupling      see also Mazur Ч Orlicz identity
Decoupling in general context      383Ч384
Decoupling in stochastic analysis      382Ч383 384
Dependence (functional)      456Ч457
Difference $\Delta^2$      2Ч3
Difference $\Delta^n$      319
Differential-space      282 288 330
Diffusion equation      284
DIM      see combinatorial dimension
DIMENSION      see also Combinatoral dimension; Hausdorfr dimension
Dimension in continuous geometries      517Ч518
Dimension in diverse contexts      517
Dimension index of interdependence      427Ч428
Dimension of 1-process      395 530Ч531
Direct products      86Ч87 see
Dissociate sets in $\Gamma$      191Ч192
Dissociate sets in $\mathbb{Z}$ and W      185
Dot product      see also Inner product in Grothendieck's inequality
Dot product, alternative representation      40 43 45Ч46 51
Dot product, Grothendieck's representation      54Ч55
Drunk's walk      331Ч333 see
Drury's theorem      190
Dual group      see Characters
Elementary tensors      see also Building blocks
Elementary tensors as characters      159
Elementary tensors as products of 'building blocks'      73 135
Elementary tensors, definition      72 80
Elementary tensors, elementary U-tensors      442 450 506
Entropy in sense of Shannon      285
Entropy metric      190
Factorization      see also Grothendieck factorization theorem; $L^2$-factorizable; Pietsch factorization theorem
Factorization of linear maps      98
Factorization of multilinear functional      103Ч105 105
Finite Fourier transform      7 see
Fourier Ч Stieltjes series      136 see
Fourier Ч Wiener series      294 see stochastic
Fractional Cartesian products      see also $\alpha$-products
Fractional Cartesian products in dimension 3/2      448
Fractional Cartesian products, definition      457Ч458
Fractional Cartesian products, preview      186Ч187 226Ч227
Fractional sum      465
Frechet measures definition (in integer dimensions)      107
Frechet measures in fractional dimensions      Chapter XII $\S5$ Chapter
Frechet measures in harmonic analysis      Chapter XIV $\S4$
Frechet variation (definition)      see also Tensor norms; variation
Frechet variation in fractional-dimensional setting      449 504
Frechet variation of function in one variable      515
Frechet variation of function in two variable      2Ч3
Frechet variation of product F-measures      255
Frechet variation of two-dimensional array      4 23
Frechet variation, $F_k$-variation in multidimensional measurable setting      111Ч112
Frechet variation, $F_k$-variation in primal setting      60
Frechet variation, 3/2-linear version      14
Frechet's theorem      see also multilinear Riesz representation theorem
Frechet's theorem, general measurable version      123
Frechet's theorem, original statement      3
Frechet's theorem, simplest multidimensional version      10
Fubini-type property, statement      67Ч68
Fubini-type property, verified by Littlewood      72
Functional independence      see also Independence
Functional independence of generalized Rademacher systems      28
Functional independence of Rademacher system      20 139 189
Functional independence, 'almost functional' independence      189
Functional independence, 1-Sidonicity      188Ч189
Functional independence, definition      187Ч188
Gauss matrix      see also finite Fourier transform
Gauss matrix , isometry      28
Gauss matrix in $\{Ч1,1\}^N$      179Ч181 258
Gauss matrix, extremal property      62
Gauss matrix, three-dimentional version      62
Gaussian distributions in Brownian movement in Einstein's model      280 284 330Ч331
Gaussian distributions in Brownian movement in Wiener's model      285Ч286 330Ч331
Gaussian distributions in Brownian movement, consequence of maximum entropy      285
Gaussian distributions in Brownian movement, simple random walk model      283Ч284
Gaussiona series      300-301 342(Exercise
Generating set      153
Gliding hump argument      72 93
Grandmasters      137 300
Grothendieck factorization theorem      see also Pietsch factorization theorem
Grothendieck factorization theorem in proof that every $F_2$-measure is protectively bounded      257
Grothendieck factorization theorem in stochastic integration      379 Chapter
Grothendieck factorization theorem, equivalent to Grothendieck's inequality      Chapter V $\S4$ 206Ч208
Grothendieck factorization theorem, multilinear extensions      Chapter V $\S5$
Grothendieck factorization theorem, statement and proof      9 96Ч97
Grothendieck inequality      see also multilinear Grothendieck inequality
Grothendieck inequality in proof of Grothendieck factorization theorem      97
Grothendieck inequality in proof that convolution is feasible      78 98Ч100
Grothendieck inequality in proof that every $F_2$-measure is protectively bounded      257
Grothendieck inequality, concise statement      39
Grothendieck inequality, constant in inequality      55 83
Grothendieck inequality, constructive proof      49
Grothendieck inequality, crucial step in 'self-contained' proofs      45
Grothendieck inequality, derivation from the Khintchin $L_1$-$L_2$ inequality (problem)      40
Grothendieck inequality, dual formulation      80 84Ч85
Grothendieck inequality, equivalent to $\Lambda(2)$-uniformizability (problem)      54
Grothendieck inequality, equivalent to Grothendieck factorization theorem      Chapter V $\S4$
Grothendieck inequality, extending to $l^p$      17 (Exercise 6)
Grothendieck inequality, extending to higher dimensions (problem)      13 88
Grothendieck inequality, formulation by Lindenstrauss and Pelczynski      8 38 87 206
Grothendieck inequality, generalization of Littlewood's mixed-norm inequality      7 11 17
Grothendieck inequality, Grothendieck's original formulation and proof      54Ч55 58 80
Grothendieck inequality, multilinear extensions      Chapter VIII
Grothendieck inequality, proofs based on $\Lambda(2)$-uniformizability      Chapter III
Grothendieck inequality, restatements      8Ч9 38Ч39 80 83 92 206 272
Grothendieck inequality, theoreme fondamental      8 9 38 45 54 88
Grothendieck inequality, two-dimensional surprise      61
Grothendieck measure of Wiener product process      404
Grothendieck measure, definition      385
Grothendieck measure, examples      387 422
Grothendieck measure, products      399
Grothendieck's 'theoreme rondamental de la theorie metrique des produits tensoriels'      8 9 38 45 54 88
Grothendieck's 'theoreme rondamental de la theorie metrique des produits tensoriels', essence      45
Grothendieck's 'theoreme rondamental de la theorie metrique des produits tensoriels', milestone      8Ч9 38 54
Grothendieck's 'theoreme rondamental de la theorie metrique des produits tensoriels', two-dimensional statement      88
Grothendieck-type inequalities      see also multilinear Grothendieck inequalities; protectively bounded Frechet measures; projectively bounded forms
Grothendieck-type inequalities in fractional dimensions      453 510
Grothendieck-type inequalities, expressed by protective boundedness      13Ч14 Chapter
Grothendieck-type inequalities, linked to product F-measures      Chapter IX $\S2$ 451Ч452
Grothendieck-type inequalities, trilinear      11 225Ч226 429
Haar measure on $\mathbb{Z}_n$      16 (Exercise 4) 34
Haar measure on $\Omega$      140Ч141
Haar measure on locally compact Abelian groups      137
Haar Ч Wiener series      342 (Exercise 13 ii) see
Hadamard sets      150
Hausdorff dimension      517 see
Hidden variables      Chapter X $\S13$ 521 see
Hilbert inequality (use by Littlewood)      6 15 62
Homogeneous chaos      311 348 see Wiener
Homogeneous integrator      362
Incidence $i_U$ (definition)      468
Indefinite stochastic integral      354 419
Independence      see also Functional independence interdependence
Independence in basic context      456Ч457
Independence in Einstein's model of Brownian motion      284 329Ч330
Independence in Wiener's model of Brownian motion      285Ч286 330Ч331
Independence, algebraic      191Ч192 436
Independence, conveyed by sub-Gaussian system      297Ч299
Independence, functional      188
Independence, heuristic sense      329
Independence, philosophical 'exercise'      342 (Exercise 14)
Independence, statistical      xvii
Independence, three notions of independence      299Ч300
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