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Goffman C., Nishiura T., Waterman D. — Homeomorphisms in Analysis
Goffman C., Nishiura T., Waterman D. — Homeomorphisms in Analysis

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Название: Homeomorphisms in Analysis

Авторы: Goffman C., Nishiura T., Waterman D.

Аннотация:

This book features the interplay of two main branches of mathematics: topology and real analysis. The material of the book is largely contained in the research publications of the authors and their students from the past 50 years. Parts of analysis are touched upon in a unique way, for example, Lebesgue measurability, Baire classes of functions, differentiability, $C^n$ and $C^{\infty}$ functions, the Blumberg theorem, bounded variation in the sense of Cesari, and various theorems on Fourier series and generalized bounded variation of a function.
Features:

Contains new results and complete proofs of some known results for the first time.

Demonstrates the wide applicability of certain basic notions and techniques in measure theory and set-theoretic topology.

Gives unified treatments of large bodies of research found in the literature.


Язык: en

Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 216

Добавлена в каталог: 02.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolute essential supremum of f (ab. ess. supf)      160 161
Absolute measure      20 159
Alexander’s horned sphere      80
Antoine’s necklace      80 84
Approximation      78
Approximation, $\varepsilon$-approximation in measure      78 79
Approximation, interpolation in measure      85 87
Area (nonparametric)      68 203 205
Area (nonparametric), area of f $(A_n(f,I))$      69
Area (nonparametric), elementary area of g $(E_n(g,I))$      68
Area (nonparametric), integral formula      68
Avdispahic      167 172
Avdispahic (theorem)      174
Baernstein      173
Baernstein — Waterman (theorem)      150
Baernstein, Waterman      152
Baire category theorem      187
Baire space      9 16 103 187
Banach      100 175
Banach Indicatrix Theorem      193
Bary      131
Berman, Brown, Cohn      49
Besicovitch      84 199
Bi-Lipschitzian      8 61 63 67 68 190
Bitopology      10
Blumberg      77
Blumberg (theorem)      9 99 104
Blumberg, property      9 104
Blumberg, set      106
Blumberg, set, simultaneous      106 108
Blumberg, set, strong      107
Blumberg, space      103 104
Bradford — Goffman (theorem)      9 99
Breckenridge, Nishiura      39
Brouwer      192
Bruckner (theorems)      8 38
Bruckner — Davies — Goffman (theorem)      17 20
Bruckner — Goffman (theorems)      21 29
Buczolich (theorem)      9 190
Category, almost continuous at x      100
Category, homogeneous second      99 102
Category, relative to E, first      99
Category, relative to E, heavy      100
Category, relative to E, second      99 100
Cell, projecting n-cell      81
Cell, topological n-cell      81 89 91 94
Cesari      38 39 70 176 177
Chanturiya (theorem)      174
Circle group $(T = \mathbb{R}/\pi\mathbb{Z})      111
Completely metrizable space      187 188
Condition GW ((h))      138—141 150
Condition UGW $((h) \& (hh))$      149 150 152 153
Continuum Hypothesis      104 160
Convolution function $(f \ast g)$      112
Convolution measure $(\mu \ast \nu)$      127
Darboux property      41 56
Davenport (lemma)      124
Deformation      90—92 95 96
Deformation, capture      93
Deformation, precise partition      93
Denjoy property $(\mu-)$      43 47
density      189
Density topology      5 23 24 63 189 195 196
Density topology $(\mathcal{T}_d)$      8 9 104
Density-closure      8 195
Density-interior      8
Direction      90 93
Dirichlet kernel $(D_n(t))$      112
Dirichlet — Jordan theorem      113 114 119 131
distribution      65
Distribution, n variables      70
Distribution, n variables, function $(f(\varphi))$      70
Distribution, n variables, measure $(\mu(\varphi))$      71
Distribution, n variables, measure representation of partial derivative      71
Distribution, n variables, partial derivative $(D_if(\varphi))$      71
Distribution, one variable, derivative $(Df(\varphi))$      66 205
Distribution, one variable, function $(f(\varphi))$      66 205
Distribution, one variable, measure $(\mu(\varphi))$      66
Distribution, one variable, measure representation of derivative      66
Essential absolute continuity      67
Essential derivative      67
Essential length      65
Essential variation      65 67 194 195
Evans, Gariepy      206
Falconer      199
Federer      188
Fejer      117
Fejer — Lebesgue theorem      112 113 117
Fourier, $P_n^R(x;f)$, $P_n^L(x;f)$      136 154
Fourier, coefficients $(a_n(f), b_n(f), \hat{f}(n))$      111 172 174
Fourier, conjugate, function $(\tilda{f})$      121
Fourier, conjugate, series $(\tilda{S}(f), \tilda{S}(x;f))$      120
Fourier, modified partial sum $(S_n^{\ast}(f), S_n^{\ast}(x;f))$      116
Fourier, multiple      177
Fourier, partial sum $(S_n(f), S_n(x;f))$      111
Fourier, series (S(f), S(x;f))      111
Frechet      205
Frechet (theorem)      205
Frechet equivalence      38
Function, n variables, approximately continuous      189 195
Function, n variables, Borel measurable      188
Function, n variables, linearly absolutely continuous (= absolutely continuous)      70
Function, n variables, linearly continuous      70
Function, n variables, Lipschitzian      68 69
Function, one variable, $\lambda$-absolutely measurable      15 159
Function, one variable, $\varphi$-bounded variation ($\varphi$-BV)      172 174
Function, one variable, absolutely continuous      62 64 193
Function, one variable, absolutely essentially bounded      20 160 161 163 180
Function, one variable, absolutely GW      164
Function, one variable, absolutely integrable      160 161 183
Function, one variable, absolutely measurable      159
Function, one variable, absolutely regulated      162—164 183
Function, one variable, approximate derivative $(f'_{ap})$      54
Function, one variable, approximately continuous      23 25 62 63
Function, one variable, approximately differentiate      78
Function, one variable, Borel measurable      188
Function, one variable, bounded deviation      178
Function, one variable, bounded variation      192
Function, one variable, generalized bounded variation      167
Function, one variable, indicatrix      131 174 194 195
Function, one variable, Lipschitzian      63
Function, one variable, quasi-continuous      104 106—108
Function, one variable, regulated      194
Function, one variable, simple      14
Function, one variable, variation $(v_f)$      21
Functions, $\lambda$-absolutely equivalent      159
Functions, absolutely equivalent      159 162 178
Functions, classes of, $G_{\log}$      175
Functions, classes of, $G_{\varphi}$      175
Functions, classes of, $\lambda$-absolutely measurable $(ab\mathcal{F})      15
Functions, classes of, Baire      188
Functions, classes of, Baire, class 0      188
Functions, classes of, Baire, class 1      188 198
Functions, classes of, Baire, class 2      188
Functions, classes of, Borel class 1      197 198
Functions, classes of, Borel measurable      188
Functions, classes of, bounded variation (BV)      192 194
Functions, classes of, bounded variation, Cesari (BVC)      70 71 176 206
Functions, classes of, continuous, bounded variation (CBV)      21 193
Functions, classes of, Darboux, Baire class 1 $(\mathcal{D}\mathcal{B}_1)$      41—43 45 47
Functions, classes of, differentiate, infinitely $(C^{\infty}[0,1])$      35
Functions, classes of, differentiate, n-times continuously $(C^n[0,1])$      30
Functions, classes of, generalized bounded variation, $V^1_{H,1}(T^2)$      177
Functions, classes of, generalized bounded variation, $V^p_{\Lambda,\alpha}(T^m)$      176 177
Functions, classes of, generalized bounded variation, $V_{\varphi}$      172—174
Functions, classes of, generalized bounded variation, $\Lambda$-bounded variation (ABV)      167 171
Functions, classes of, generalized bounded variation, harmonic bounded variation (HBV)      168—171 173—175 177
Functions, classes of, generalized bounded variation, V[h]      173 178
Functions, classes of, GW      169
Functions, classes of, Lipschitz class $\alpha(\Lambda_{\alpha})$      114
Functions, classes of, regulated (R(T))      113 136 138 139 141 175
Functions, classes of, UGW      169
Functions, classes of, Zahorski $(\mathcal{M}_0)$      41 42
Functions, classes of, Zahorski $(\mathcal{M}_1)$      41 42
Functions, complementary      173 176
Garsia, Sawyer      175
Gleyzal      44
Gleyzal (theorem)      44 54
Goffman      70 78 80 88 168—170 173 176
Goffman (theorems)      71 79 83 84 86—88
Goffman — Liu (theorem)      71
Goffman — Neugebauer (theorem)      56
Goffman — Waterman (theorems)      138 177 196
Goffman, Neugebauer, Nishiura      195
Goffman, Pedrick      97
Goffman, Waterman      77 117 196 198
Goffman, Zink      77
Gorman, decomposition      5
Gorman, example      19
Gorman, sequence      5
Gorman, theorem      7 14 20
hadamard      178
Hardy      122
Haupt, Pauc      195
Hausdorff dimension, $dim_H E$      199
Hausdorff measure      68 97 104
Hausdorff measure, $H_1(v_f[K_f])$      30
Hausdorff measure, $H_n(E)$      71
Hausdorff measure, $H_{\alpha}(E)$      71 199
Hausdorff measure, $H_{\alpha}^{\delta} (E)$      199
Hausdorff measure, $\delta$-cover      198
Hausdorff measure, counting      199
Hausdorff measure, outer      199
Hausdorff naive packing, $npH_1(v_f[K_f])$      30
Hausdorff naive packing, $npH_{\alpha}(E)$      200
Hausdorff naive packing, $npH_{\alpha}(v_f[K_f])$      37 38
Hausdorff naive packing, $npH_{\alpha}^{\delta}(E)$      200
Hausdorff naive packing, $npH_{\frac1n}(v_f[K_f])$      30
Hausdorff naive packing, $\delta$-packing      199
Hausdorff naive packing, $\Gamma_{\alpha}(E)$      200
Hausdorff naive packing, rarefaction index $(\Lambda(E))$      202
Hilbert transform (H[f])      122
Homotopy      89 94 95 98
Integral formula, area      68 69
Integral formula, length      63 64
Integral mean value theorem      141
Interval function      44 47 54
Interval function (convergence of)      44
Interval function f([a, b]) = f(b) — f(a)      131 138
Jacobian (J(h, x))      68
JORDAN      64
Jordan — Schoenflies (theorem)      80 81 84
Jurkat — Waterman (theorems)      121
Jurkat, Waterman      120 124
Kahane — Katznelson (theorem)      117
Kahane, Katznelson      117 124
Kahane, Zygmund      131
Laczkovich — Preiss (theorem)      31 37
Lebesgue      ix 68 204
Lebesgue equivalence      3 21 65
Length (nonparametric)      63 203
Length (nonparametric), elementary length of g $(E_1(g, [0,1]))$      63 203
Length (nonparametric), essential length of f (ess l(f, [0,1]))      63 203 length
Levy      103
Lower semicontinuous functional      64
Lower semicontinuous functional of n variables functions, area      69
Lower semicontinuous functional of n variables functions, elementary area      69 204
Lower semicontinuous functional of n variables functions, variation $(V_i(f, I), i = 1, 2,... , n)$      69
Lower semicontinuous of one variable functions, elementary length      64 204
Lower semicontinuous of one variable functions, essential variation      194
Lower semicontinuous of one variable functions, length      64
Lower semicontinuous of one variable functions, total variation      64 193
Lukes, Maly, Zajicek      10
Lusin (theorem)      77 78 84 86 104
Map, approximately continuous      197
Map, bi-Lipschitzian      189 190
Map, dilation      190
Map, Lipschitzian      189 190 199
Map, Lipschitzian, constant of f (Lip(f))      189
Maps, classes of, bimeasurable $(\mathcal{M}[I^n,I^m])$      85—87
Maps, classes of, bimeasurable $(\mathcal{M}[I^n,I^m])$, interpolation metric      85
Maps, classes of, self-homeomorphism (H[In])      85—87
Maximoff (theorem)      41 56
Mean, (C,l)      112
Mean, Abel      113 121
Measurable boundary (upper, lower)      78
Measure, $(\mu,\nu)$-admissible      95
Measure, equivalence      43
Measure, Lebesgue equivalence      89 98
Measure, Lebesgue — Stieltjes      91
Measure, Lebesgue-like      43 46 47 89 90 93 95
Measure, positive      43
Modulus of continuity $(\omega(\delta;f))$      114
Modulus of continuity, integral $(\omega_p(\delta;f))$      115
Neugebauer      43 57 104 106
Neugebauer (theorems)      45 46 108
Olevskii (theorem)      124 129
Oniani      121
Oscillation of f on E (osc(f; ?))      162
Oskolkov      173
Oxtoby      15 20 97 160
Oxtoby, Ulam      97
Pal — Bohr theorem      xii 111 117 120 121
Partition, boundary collection      94
Partition, mesh      94
Partition, n-cell      94
Partition, precise partition      95
Pierce — Waterman theorem      139
Point of almost continuity      see “Category almost
Point of almost continuity, density      8 63 189 190 195
Point of almost continuity, dispersion      8 63 189 190
Point of almost continuity, varying monotonicity      27
Preiss      41 47 56
Preiss (theorem)      43
Prus — Wisniowski      176
Radon — Nykodym derivative $(d\mu = gdx)$      67
Rarefaction index      202
Rarefaction index, $\Lambda(v_f,[K_f])$      36
Representation      38
Riemann localization principle      177
Riemann mapping theorem      117 122
Riesz, F. and M.      122
Rogers      198
Saakyan      117 121
Saks — Sierpiriski (theorem)      77 78
Salem      117 131
Salem, theorem      132
Schramm, Waterman      172
Schwarz, example      203
Selection      43 46 47
Set, $\eta_1$-set      103
Set, $\lambda$-absolute measure 0      15 159
Set, $\lambda$-absolutely measurable      15 159
Set, absolutely measurable      159
Set, analytic (= Suslin)      3 15 188
Set, Blumberg      see “Blumberg set”
Set, Borel, $F_{\sigma}$      188 195
Set, Borel, $G_{\sigma}$      188
Set, c-set      5
Set, closure of $E (\bar{E})$      187
Set, connected      196
Set, continuity set of f (C(f))      106
Set, cozero-set      195
Set, density-closed      195
Set, density-open      5 195
Set, homogeneous second category      99
Set, hyper plane      90 91
Set, interior of $E (E^0)$      187
Set, k-flat      90
Set, porous      4 62 189
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