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

Главная Ex Libris Книги Журналы Статьи Серии Каталог Wanted Загрузка ХудЛит Справка Поиск по индексам Поиск Форум

Авторизация

Поиск по указателям

Korevaar J. — Tauberian Theory: A Century of Developments

Обсудите книгу на научном форуме

Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter

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.

Язык:

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

Matrix method, limitable sequences      258
Matrix method, perfect      259 261
Matrix method, regularity criterion      258
Matrix method, reversible      258
Matrix method, sectional convergence      261
Maximal ideal      237 241
Maximal ideal in $A_{\omega}$ , characterization      244 247
Maximal ideal space      243
Maximal ideal, (closed) in $L^{1}$       237
Maximal ideal, (closed) in $L_{\omega}$       250
Maximal ideal, Gelfand — Mazur theorem      241
Maximum principle for subharmonic function      367
Maximum principle, Phragmen — Lindeloef theorem      376
Measure from function of bounded variation      25
Measure, Laplace transform of      26
Measure, variation measure      25
Mellin convolution      197
Mellin convolution, Mercerian theorem      203
Mellin transform      94 99 122 195
Mellin transform, complex Abelian theorem      123
Mellin transform, complex Tauberian      124
Mellin transform, Wiener — Ikehara theorem      124
Mellin — Stieltjes convolution      195
Mellin — Stieltjes convolution, Abelian theorem      195
Mellin — Stieltjes convolution, Tauberian theorem      196
Mellin — Stieltjes transform      122 see
Melnik and Gaier, Borel high-indices theorem      280 312
Mercer's theorem      202
Mercerian theorem      202
Mercerian theorem in higher-order theory      204
Mercerian theorem, Drasin — Shea theorem      204
Mercerian theorem, involving Laplace transform      203
Mercerian theorem, involving Mellin convolution      203
Mercerian theorem, involving regular variation      203
Mercerian theorem, involving Stieltjes transform      203
Meyer — Koenig and Zeller      see also "Functional-analytic method"
Meyer — Koenig and Zeller, boundedness principle      262
Meyer — Koenig and Zeller, high-indices theorems      270 280 312 334
Meyer — Koenig, circle methods      280 290 306
Milin's theorem      274
Milloux problem      367
Minimal ideal      238 248
Minimal ideal, belonging to $\infty$       248
Mittag-Leffler star      285
Moebius function      8 134
Moment theorem      77
Monotone minorant      41 280
Monotone minorant for boundedness theorem      40 43 296
Muentz — Szasz formula, theorem      318 335
Newman contour integration method      119 133
Newman proof of PNT      134
Non-continuability, Fabry theorem      294
Non-continuability, Hadamard theorem      280 294
Norm of polynomial involving coefficients      356 361
One-sided $L^{1}$ approximation by entire functions      128
One-sided $L^{1}$ approximation by polynomials      24 352 356 364
One-sided $L^{1}$ approximation of smooth function      362
One-sided $L^{1}$ approximation, Graham — Vaaler construction      128
One-sided $L^{1}$ approximation, Markov — Stieltjes construction      359
Open problem involving, Beurling slow variation      200
Open problem involving, Beurling's approximation theorem      241
Open problem involving, convolution of sequences      409
Open problem involving, Erdos' nonlinear Tauberian      412
Open problem involving, lacunary power series      324
Open problem involving, Laplace — Stieltjes transform      350
Open problem involving, Littlewood's theorem      173
Open problem involving, recurrent events      273
Open problem involving, Wiener — Ikehara theorem      134
Operator theory, complex Tauberian      163—165
Operator theory, motivating Fatou — Riesz extensions      119 150 154
Operator theory, power bounded operator      163
Operator theory, resolvent set      163
Operator theory, semigroup of operators      165
Operator theory, spectrum      163
Optimality of Hadamard gaps      56
Optimality of remainder estimates      369
Optimality of square-root gaps      322
Optimality of Tauberian condition      15 54 56 57 103 106 320
Ostrowski gaps      292
Ostrowski theorem      280 294
Overconvergence      288 293
Paley — Wiener theorem      128
Paley — Wiener theorem, another P-W theorem      246
Partitions      179 208 217
Partitions of positive integer      228
Partitions, generating function      229
Partitions, strong asymptotics      228
Pathology related to limit relation of slow variation      184
Peak function in Tauberian theory      2 14 50 351
Peak function, integrated peak function, or ramp function      50
Periodic distribution      155
Periodic distribution, Fourier series      155
Periodic distribution, pseudofunction      156
Periodic distribution, pseudomeasure      156
Phragmen — Lindelof theorem      376 377
Pitt      see also "Wiener — Pitt"
Pitt, Borel high-indices theorem      280 312
Pitt, boundedness theorem      235 250 316
PNT      see "Prime number theorem"
Poisson integral      7
Poisson integral for half-space      112
Poisson integral of positive measure      111
Poisson kernel      7
Poisson kernel for half-space      112
Polya (Polya) peaks      207
Polynomial approximation in Karamata's proof      21
Polynomial approximation in remainder theory      352
Polynomial approximation in Wielandt's proof      23
Polynomial approximation, monotonic approximation to unit step function      50
Polynomial approximation, one-sided $L^{1}$ approximation      352 356
Polynomial approximation, Weierstrass theorem      21 77
Power bounded operator      163
Power series method $J_{\mu}$       335
Power series method $J_{\mu}$ , Abel-type      336
Power series method $J_{\mu}$ , Borel-type      336
Power series method $J_{\mu}$ , Tauberian theorem      337
Power series, Abelian theorem      16
Power series, Hardy — Littlewood theorem      15 16
Power series, Littlewood's theorem      14
Power series, Tauber's theorems      10 11
Primary ideal      250
Prime number theorem      19 118
Prime number theorem and zeros of $\zeta(z)$       20 124
Prime number theorem, derivation using, Ingham summability      111
Prime number theorem, derivation using, Lambert Tauberian      19 93
Prime number theorem, derivation using, Landau Tauberian theorem      121
Prime number theorem, derivation using, Landau — Ingham asymptotics      110
Prime number theorem, derivation using, Newman method      134
Prime number theorem, derivation using, Newman type method      140
Prime number theorem, derivation using, Selberg relation      402
Prime number theorem, derivation using, Wiener — Ikehara theorem      122
Prime number theorem, methods of proof      118 119 134
Proper ideal, contained in maximal ideal      237
Pseudofunction      138 148 156
Pseudofunction, local pseudofunction behavior      157
Pseudomeasure      156
Quadrature theory, Christoffel numbers      361
Quasi-analyticity theorem      247
Ramp function, integrated peak function      50
Ratio theorem for Laplace transform      197
Recurrent events      270
Recurrent events, generating functions      271
Recurrent events, mean recurrence time      271
Recurrent events, open problem      273
Regular Banach algebra      246
Regular summability method      3 258
Regular variation      179
Regular variation in probability theory      178
Regular variation, background      177
Regular variation, book BGT      178

Regular variation, characterization involving integrals      187
Regular variation, class $RV_{\alpha}$       180
Regular variation, de Haan theory      189 200
Regular variation, Feller's characterization      181
Regular variation, general-kernel theorem      195 196
Regular variation, higher-order theory      189 200
Regular variation, highlights      179
Regular variation, Mercerian theorem      203
Regular variation, O-regular variation      191 197
Regular variation, related concepts      188
Regularly varying function      178 180
Regularly varying function in integral      186—188
Regularly varying function, characteristic limit relation      181
Regularly varying function, index      180
Regularly varying function, monotonic equivalent      180
Regularly varying function, smooth equivalent      181
Remainder estimate involving, Dirichlet series      373
Remainder estimate involving, high-indices      374
Remainder estimate involving, Lambert series      400
Remainder estimate involving, Laplace transform      347 352
Remainder estimate involving, Laplace transform, alternative form      348
Remainder estimate involving, Laplace — Stieltjes transform      351 381 397
Remainder estimate involving, Laplace — Stieltjes transform, Freud theorem      353
Remainder estimate involving, Laplace — Stieltjes transform, Ganelius theorem      349
Remainder estimate involving, power series      348
Remainder estimate involving, Riccati equation      412
Remainder estimate involving, Stieltjes transform      392
Remainder estimate involving, Wiener kernel      379 381 389 394
Remainder estimates, optimality      369
Remainder theory      343 see
Remainder theory for Wiener theorem      398
Remainder theory, convolution of sequences      400
Remainder theory, convolution of sequences, Erdos problem      401
Remainder theory, convolution of sequences, square-root problem      401
Remainder theory, critical rate      350 393
Remainder theory, Fourier integral method      344
Remainder theory, Fourier integral method, analyticity of kernel      379
Remainder theory, Fourier integral method, conditions on derivatives      398
Remainder theory, Fourier integral method, history      379
Remainder theory, Fourier integral method, model theorem      381
Remainder theory, Freud condition      349
Remainder theory, Ganelius' Fourier method      381
Remainder theory, involving Abel summability      343 380
Remainder theory, involving Abel summability, history      344
Remainder theory, involving Cesaro summability      14
Remainder theory, Lyttkens' Fourier method      398
Remainder theory, Lyttkens' Fourier method, example      399
Remainder theory, model result one      345
Remainder theory, model result two      346
Remainder theory, model result two, extension      347
Remainder theory, polynomial approximation method      343 346
Remainder theory, vanishing remainder      350 352 366 374
Renewal theory      236
Renewal theory, Blackwell      98 271
Renewal theory, Erdos, Feller and Pollard      271
Repeated differentiation method      2 14 34
Residue theorem      377
Resolvent set      163
Restricted high-indices theorem      294 311
Restricted high-indices theorem for general circle method      312 313
Restricted high-indices theorem for special circle methods      334
Riccati equation      412
Riemann hypothesis      64 400
Riemann localization principle      148 162
Riemann summability      79
Riemann zeta function      see "Zeta function"
Riemann — Lebesgue lemma      79 80 121 126 131 137 145 150 152 237
Riesz, extension of Fatou theorem      118 150
Riesz, mean      39 308 341
Riesz, summability      307 308
Schmidt condition      32 33 40 279 295
Schmidt condition for Abel summability      15
Schmidt condition for Borel summability      18 305
Schmidt condition, optimality      57 104
Schwartz space in distribution theory      91
Sectional convergence      261
Selberg, (and Erdos) PNT      63 118
Selberg, one-sided $L^{1}$ approximation      128
Selberg, relation in prime number theory      402
Selection principle      14 71 192 204
Self-neglecting function      200 233
Semigroup of operators      165
Seminorms      91 258
Seminorms, separating family      258
Sequence space      257
Sequence space, bounded, convergent, divergent vector      258
Sequence space, eventually constant vector      262
Sequence space, sectional convergence      261
Set of smallness for bounded analytic function      366
Siegel, fundamental relation      403 412
Skeleton, of ideal      248
Sliding hump method      257 264
Slowly decreasing function on $\mathbb{R}$       69 142
Slowly decreasing function on $\mathbb{R}^{+}$       33 74
Slowly decreasing partial sums      14 15 19
Slowly decreasing, very slowly decreasing      143 149
Slowly oscillating function on $\mathbb{R}$       69
Slowly oscillating function on $\mathbb{R}^{+}$       32 68
Slowly oscillating partial sums      15
Slowly varying function      32 167 178 179
Slowly varying function in integral      186
Slowly varying function in ratio theorem      197
Slowly varying function, additive equivalent      181
Slowly varying function, additive equivalent, integral representation      182
Slowly varying function, basic properties      180
Slowly varying function, Beurling slow variation      199
Slowly varying function, boundedness      180
Slowly varying function, complex-valued      184
Slowly varying function, inequalities      180
Slowly varying function, integral representation      180
Slowly varying function, limit relation      180
Slowly varying function, limit relation, possible pathology      184
Slowly varying function, limit relation, uniform convergence      180
Slowly varying function, logarithm      181 188
Slowly varying function, related concepts      188
Slowly varying function, very slow variation      185
Spectrum      163
Spectrum, continuous spectrum      163
Spectrum, peripheral spectrum      163
Spectrum, point spectrum      163
Spectrum, residual spectrum      163
Square-root gaps      270 311
Square-root gaps, growth of power series with      323
Square-root gaps, optimality in high-indices theorem      322
Square-root problem      401
Square-root problem, open problem      409
Square-root problem, remainder theorem      403
Statistical convergence      57
Step function condition for $\mathbb{R}$       69 143
Step function condition for $\mathbb{R}^{+}$       33
Stieltjes integral from series      24 26
Stieltjes series      101
Stieltjes transform of first order      46
Stieltjes transform, approximate inversion formula      174
Stieltjes transform, counting eigenvalues      175
Stieltjes transform, counting zeros      175
Stieltjes transform, Karamata's theorem      194
Stieltjes transform, Mercerian theorem      203
Stieltjes transform, remainder theory      173 392
Stieltjes transform, remainder theory, complex theory      173
Stieltjes transform, Tauberian theorem      47
Stirling's formula      10 107 289 318 337
Strong asymptotic equivalence      220
Strong asymptotics      208 212 218 228
Strong asymptotics, model results      214 217
Subharmonic function, maximum principle      367
Summability method(s), (A, $\lambda$ ) method      48
Summability method(s), Abel      1 4
Summability method(s), Abel — Poisson      1
Summability method(s), and probability      292 309

1 2 3 4

О проекте