Korevaar J. — Tauberian Theory: A Century of Developments :: Электронная библиотека попечительского совета мехмата МГУ

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Korevaar J. — Tauberian Theory: A Century of Developments

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Название: Tauberian Theory: A Century of Developments

Автор: Korevaar J.

Аннотация:

Tauberian theory compares summability methods for series and integrals, helps to decide when there is convergence, and provides asymptotic and remainder estimates. The author shows the development of the theory from the beginning and his expert commentary evokes the excitement surrounding the early results. He shows the fascination of the difficult Hardy-Littlewood theorems and of an unexpected simple proof, and extolls Wiener's breakthrough based on Fourier theory.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 483

Добавлена в каталог: 05.12.2009

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Предметный указатель

Fatou theorem for harmonic function, converse theorem      113 115
Fatou theorem, complex Tauberian      118 148
Fatou theorem, extensions      149—151 153 157—159
Fatou theorem, remainder      167
Fatou theorem, with one-sided condition      150
Fatou — Riesz theorem      149 150
Fejer kernel      6
Fejer kernel for $\mathbb{R}$       80 125 254
Fejer kernel, related kernels      135 167
Fejer's theorem for Fourier series      6
Fejer's theorem Tauberian theorem      12 276
Finite differences      35
Finite differences, calculus of      325
Finite form of (Laplace) complex Tauberian      138 162
Finite form of Wiener — Ikehara theorem      128
Fourier series of periodic distribution      155
Fourier series, Abel summability      7
Fourier series, Cesaro summability      6
Fourier series, convergence by Tauberian theory      13
Fourier series, Fejer's theorem      6
Fourier series, Wiener's lemma      66 84 243
Fourier Tauberian theorems      176
Fourier transformation      69 79
Fourier transformation in remainder theory      379
Fourier transformation of convolution      80
Fourier transformation of tempered distribution      91
Fourier transformation of testing equation      80
Fourier transformation of Wiener kernel      81
Fourier transformation, Berry — Esseen inequality      232
Fourier transformation, complex Fourier transform      240
Fourier transformation, important Fourier pair      80
Fourier — Mellin transform      81 84 94 108 110
Frechet coordinate space FK-space      257 258
Frechet coordinate space, (Banach) BK-space      262
Frechet coordinate space, basis      261
Frechet coordinate space, boundedness principle      257 260 262
Frechet coordinate space, sectional convergence      261
Freud condition      349
Freud, remainder theory      343 345 353
Frobenius' continuity theorem      117
Frobenius' continuity theorem, analog for Dirichlet series      118
Fubini's theorem      80 82
Functional-analytic method, FK-space      235 257
Fundamental relation of Siegel, in Erdos' convolution problem      403 412
Fundamental subset      259
Gaier and Melnik, Borel high-indices theorem      280 312
Gamma function, growth of reciprocal      381
Ganelius, remainder theory      349
Ganelius, remainder theory, Fourier method      380
Ganelius, remainder theory, inequality      383
Ganelius, remainder theory, model theorem      381
Ganelius, remainder theory, Wiener theorem      398
Gap condition      149
Gap condition, general      266
Gap condition, Hadamard gaps, high indices      50
Gap condition, Ostrowski gaps      292
Gap condition, square-root gaps      280 311
Gap series, (C,k) summability      14
Gap series, boundedness theorem      50 52
Gap series, high-indices theorem      50
Gap Tauberian theorem      266 267
Gap Tauberian theorem for Abel summability      269
Gap Tauberian theorem for Borel summability      312
Gap Tauberian theorem for Cesaro summability      269
Gap Tauberian theorem for Euler summability      270
Gap Tauberian theorem of Ingham      150
Gap theorem of Fabry      294
Gap theorem of Gaier      317 318
Gap theorem of Hadamard      294
Gap-perfect method      267
Gap-perfect method, Abel summability      268
Gap-perfect method, Cesaro summability      267
Gauss summability of integral      61
Gelfand isometry theorem      164
Gelfand representation      243
Gelfand theory      241
Gelfand transform      243 245
Gelfand — Mazur theorem      241
General circle method      281
General circle method and normal distribution      292
General circle method, basic properties      288
General circle method, basic properties, consequences      290
General circle method, boundedness theorem      295
General circle method, definition      290
General circle method, integral formula for limitability      297 299—301
General circle method, restricted high-indices theorem      312 313
General Dirichlet series      see "Dirichlet series"
General-kernel theorem      195 196 202
General-kernel transform      25 100
General-kernel transform, boundedness theorem      43
General-kernel transform, integration by parts      44
Generating function, partitions      229
Generating function, recurrent events      271
Generating function, square-root problem      404 408
Growth of lacunary series      323
Growth of lacunary series, open problem      324
Growth of lacunary series, with square-root gaps      323
Hadamard gaps      50 269 294
Hadamard gaps, optimality in high-indices theorem      56
Hadamard non-continuability theorem      280 294
Hadamard PNT      19 63 118
Hadamard sequence      33 50 323 376
Hahn — Banach theorem, continuous linear functionals test      74 249
Hamel basis      185
Hardy book Divergent Series      1 3
Hardy class $H^{1}$       150
Hardy class $H^{1}$ , local $H^{1}$ condition      151
Hardy — Landau theorem for Cesaro summability      12
Hardy — Littlewood theorem for Abel summability      15
Hardy — Littlewood theorem for Abel summability by Wiener theory      77 95 101
Hardy — Littlewood theorem for Abel to Cesaro summability      16 56
Hardy — Littlewood theorem for Abel to Cesaro summability by Wiener theory      77
Hardy — Littlewood theorem for Borel summability      18 279 302
Hardy — Littlewood theorem for Lambert summability      18
Hardy — Littlewood theorem for Lambert summability by Wiener theory      92 101
Hardy — Littlewood theorem, high-indices theorem      2 18 50
Hardy — Littlewood theorem, involving Laplace transform      29
Hardy — Littlewood theorem, involving power series      15 16 178
Hardy — Littlewood theorem, involving power series by Wiener theory      103
Hardy — Littlewood theorem, involving power series, relaxed Tauberian condition      349
Hardy — Littlewood theorem, involving Stieltjes transform      47
Hardy — Littlewood theorem, Karamata's method      20
Hardy — Littlewood theorem, Karamata's method, Wielandt form      22
Hardy — Littlewood theorem, remainder theory for      169 345—349
Hardy — Littlewood theorem, remainder theory for, complex theory      170
Harmonic function, Herglotz representation      111
Harmonic function, Poisson integral      111
Harmonic measure      367
Harmonic-measure argument      366 376
Hausdorff maximality theorem      237
Hayman regularity theorem      274
Heaviside function, unit step function      156
High-indices theorem      2 18 50 102 269 270 311
High-indices theorem for Borel summability      280 312
High-indices theorem for Euler summability      312
High-indices theorem for Lambert summability      54
High-indices theorem for Taylor method      334
High-indices theorem, Ingham method of proof      50
High-indices theorem, method of Halasz      375
High-indices theorem, related theorems      54
High-indices theorem, remainder estimate      374
High-indices theorem, restricted high-indices theorem      281 311
Higher-order theory in regular variation      189 200
Higher-order theory in regular variation, Mercerian theorem      204
Hoelder summability      4
Homomorphism      242
Homomorphism, kernel      242
Hull, of ideal      248
Ideal      237
Ideal, belonging to $\infty$       237 248

Ideal, belonging to set S      248
Ideal, closed ideal in $L^{1}$       238
Ideal, hull or skeleton      248
Ideal, maximal      237
Ideal, minimal      238 248
Ideal, primary      250
Ideal, proper      237
Ikehara's theorem      see "Wiener — Ikehara theorem"
Improper Stieltjes integral, integration by parts      27
Improper Stieltjes integral, notation      25
Index of regularly varying function      180
Inequalities, for slowly varying function      167 180
Inequality of Berry — Esseen      232 384
Inequality of Bieberbach — de Branges      274
Inequality of Ganelius      383
Inequality of Lebedev — Milin      274
Ingham summability      110
Ingham summability, Tauberian theorem      110
Ingham, asymptotics for partitions      228 229
Ingham, Borel high-indices theorem      312
Ingham, complex Tauberian      133 135
Ingham, gap Tauberian theorem      150
Ingham, Landau — Ingham asymptotics      106
Ingham, peak function      50 351
Ingham, strong complex Tauberian      222
Integral formulas for limitability by circle method      297 300
Integral representation for slowly varying function      180
Integral, (C,k) summable      38
Integral, Abel summable      26
Integral, Cesaro summable      26
Integral, improper integral, notation      25
Integral, Lambert summable      26
Integral, Lebesgue — Stieltjes      25
Integration by parts in general-kernel transform      44
Integration by parts in Lambert transform      28
Integration by parts in Laplace — Stieltjes transform      28
Isometry, Gelfand theorem      164
Jackson kernel      346
Jackson kernel, for $\mathbb{R}$       135
Jensen's theorem, for bounded analytic function      377
Karamata method for Hardy — Littlewood theorem      2 14 20 31
Karamata method for Hardy — Littlewood theorem, Wielandt form      22
Karamata regular variation      178
Karamata regular variation, characterization      186
Karamata regular variation, definition      179
Karamata's theorem involving, Laplace — Stieltjes transform      192
Karamata's theorem involving, Stieltjes transform      194
Kernel of homomorphism      242
Kernel, well-approximated by difference kernel      251 314
Kloosterman's method for Cesaro Tauberian      13
Kratz and Stadtmueller, Tauberians for power series methods      337 338
Lambert summability      2 5 18 25 67
Lambert summability in number theory      8 9
Lambert summability of integral      26
Lambert summability to Abel summability      20
Lambert summability, Abelian theorem      5
Lambert summability, high-indices theorem      54
Lambert Tauberian      18 65 93 101
Lambert Tauberian to prime number theorem      19
Lambert, kernel      5
Lambert, series      5 9
Lambert, sum      5 9 10 140
Lambert, transform      26 94
Lambert, transform, Tauberian theorem for      94
Landau — Ikehara theorem      see "Wiener — Ikehara theorem"
Landau — Ingham asymptotics      107
Landau — Ingham asymptotics in proof of PNT      110
Landau, Cesaro Tauberian      12
Landau, complex Tauberian      120
Laplace asymptotic method      211 220
Laplace transform      192
Laplace transform of exponential growth      208
Laplace transform of exponential growth, strong asymptotics      214
Laplace transform of measure      26
Laplace transform of rapid growth      207 209
Laplace transform of rapid growth, complex approach      222
Laplace transform of rapid growth, logarithmic theory      208
Laplace transform of rapid growth, Martin — Wiener asymptotics      212
Laplace transform of rapid growth, strong asymptotics      208 212
Laplace transform of very rapid growth      217
Laplace transform of very rapid growth, logarithmic theory      219 221
Laplace transform of very rapid growth, strong asymptotics      218
Laplace transform, complex Tauberian      124 135 141 142 154 161
Laplace transform, contour integration method      136
Laplace transform, from power series      26 347
Laplace transform, Hardy — Littlewood type theorem      29 46
Laplace transform, log-convex      233
Laplace transform, ratio theorem      197
Laplace transform, remainder estimate      347 352
Laplace transform, remainder estimate, optimality      371
Laplace transform, Tauberian theorem      28 30
Laplace transform, two-sided      232
Laplace transform, Wiener — Ikehara theorem      124
Laplace — Stieltjes transform      26 118 192 380 see
Laplace — Stieltjes transform, Continuity Theorem      192
Laplace — Stieltjes transform, extension of Tauber's theorems      28
Laplace — Stieltjes transform, half-open problem      350
Laplace — Stieltjes transform, integration by parts      28
Laplace — Stieltjes transform, Karamata's theorem      193
Laplace — Stieltjes transform, Mercerian theorem      203
Laplace — Stieltjes transform, remainder estimate      349 351 353 381 397
Laplace — Stieltjes transform, remainder estimate for Cesaro means      350 354
Laplace — Stieltjes transform, standard conditions      345
Laplace — Stieltjes transform, Tauberian theorem      30 46
Large deviations      208 292
Lebedev — Milin inequalities      274
Lebesgue — Stieltjes integral      25 98
Limit relation for slowly varying function      180
Limit relation, characterizing regularly varying function      181
Limitability by general circle method      290
Limitability of function      26
Limitability of sequence      3
Limitation domain      258
Limitation kernel      65 69 73
Liouville theorem      242
Little o-condition      see "Tauberian condition"
Littlewood's condition      2
Littlewood's condition, corresponding one-sided condition      15
Littlewood's theorem      2 14
Littlewood's theorem by Wiener theory      77
Littlewood's theorem, Karamata's proof      22
Littlewood's theorem, open problem      173
Littlewood's theorem, optimal character      54
Littlewood's theorem, remainder theory for      346 369
Littlewood's theorem, remainder theory for, complex theory      172
Littlewood's theorem, Schmidt's extension      43
Littlewood's theorem, Schmidt's extension, further extension      46
Littlewood's theorem, Wielandt's proof      22 346
Littlewood, proof of PNT      118
Localization principle of Riemann      148
Log-concave density as Tauberian condition      234
Log-concave density of positive measure      233
Log-convex, Laplace transform      233
Logarithmic asymptotics      208 229
Logarithmic asymptotics, model results      209 210 219
Logarithmic asymptotics, more general result      220
Logarithmic integral, li      218
Logarithmic summability      336
Logarithmico-exponential function      177
Majorant of type $\lambda$       128 129
Majorant, extremal      129
Mangoldt (von) function      9
Mangoldt (von), counting prime factors      310
Martin — Wiener asymptotics for Laplace transform      212 214
Mass function      192
Matrix method      258
Matrix method, conservative      258
Matrix method, domain of definition      258
Matrix method, effective or limitation domain      258
Matrix method, gap-perfect      267

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