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

Summability method(s), arithmetic means      1
Summability method(s), Borel method(s)      5 279
Summability method(s), Cesaro (C,k)      3
Summability method(s), Cesaro or (C,1)      1 3
Summability method(s), circle methods      280 328
Summability method(s), comparison      3 4 58
Summability method(s), conservative      258
Summability method(s), consistent      3
Summability method(s), equivalent      4
Summability method(s), Euler      326
Summability method(s), exponential method      279
Summability method(s), Gauss (for integral)      61
Summability method(s), general circle method      281 290
Summability method(s), generalized Borel      331
Summability method(s), Hoelder (H,k)      4
Summability method(s), Ingham      110
Summability method(s), Lambert      5
Summability method(s), logarithmic summability      336
Summability method(s), matrix methods      258
Summability method(s), method $S_{\alpha}$       330
Summability method(s), methods (E,q), $E_{\alpha}$       328
Summability method(s), methods B and B'      282
Summability method(s), methods B and B', near equivalence      282
Summability method(s), Poisson — Abel      7
Summability method(s), power series method (the)      1
Summability method(s), power series methods      335
Summability method(s), regular      3 258 336
Summability method(s), Riesz      307 308
Summability method(s), statistical convergence      57
Summability method(s), stronger      3
Summability method(s), Taylor method      280 329
Summability method(s), Valiron method      280 331
Summability method(s), weighted-mean method      341
Symmetric derivative      112
Szasz, Muntz — Szasz theorem      318 335
Szasz, Tauberian conditions      15 49
Tauber's theorem for power series      10
Tauber's theorem for power series, second theorem      11
Tauber's theorem, extension to Laplace transforms      28
Tauberian (as noun)      see "Tauberian theorem"
Tauberian condition      2 6
Tauberian condition (complex theory), boundedness condition      133 136
Tauberian condition (complex theory), Fatou condition      118 148
Tauberian condition (complex theory), gap condition      149
Tauberian condition (complex theory), positivity condition      120 122 124 139 141
Tauberian condition (complex theory), slow decrease      142
Tauberian condition (complex theory), step function condition      143
Tauberian condition (complex theory), very slow decrease      143 149
Tauberian condition for Abel summability      12 14
Tauberian condition for Borel summability      18 279
Tauberian condition for Cesaro but not Abel summability      57
Tauberian condition for Cesaro summability      12
Tauberian condition for Dirichlet series      17 49
Tauberian condition for general circle method      305
Tauberian condition for method $J_{\mu}$       337
Tauberian condition, area condition of conformal mapping      12
Tauberian condition, average-type condition      15
Tauberian condition, average-type condition for Abel to Cesaro summability      58
Tauberian condition, average-type condition for general circle method      321 333
Tauberian condition, big $\mathcal{O}$ -condition      12 14
Tauberian condition, gap condition      50 266
Tauberian condition, little o-condition      12 58 337
Tauberian condition, Littlewood's condition      2 14
Tauberian condition, log-concave density      234
Tauberian condition, optimality      15 54 56 57 103 106 320
Tauberian condition, relation with probability      281
Tauberian condition, relaxation      58
Tauberian condition, Schmidt condition      32 33 295
Tauberian condition, Schmidt condition for Abel summability      15
Tauberian condition, Schmidt condition for Borel summability      18
Tauberian condition, slow decrease of partial sums      14 15 19
Tauberian condition, slow decrease on $\mathbb{R}$       69
Tauberian condition, slow decrease on $\mathbb{R}^{+}$       33 74
Tauberian condition, slow oscillation of partial sums      15
Tauberian condition, slow oscillation on $\mathbb{R}$       69
Tauberian condition, slow oscillation on $\mathbb{R}^{+}$       32 68
Tauberian condition, step function condition for $\mathbb{R}$       69
Tauberian condition, step function condition for $\mathbb{R}^{+}$       33
Tauberian condition, Szasz conditions      15 49
Tauberian condition, Valiron condition      295 302
Tauberian condition, Wiener conditions      95
Tauberian constant      11
Tauberian operator      262
Tauberian problem      6 67
Tauberian problem, Wiener — Pitt question      68 69
Tauberian problem, Wiener's question      73
Tauberian theorem      6 see
Tauberian theorem for (C,k) summability      14
Tauberian theorem for (C,k) summability of integral      39
Tauberian theorem for Abel summability      10 11 14 15
Tauberian theorem for Borel summability      18 302 305
Tauberian theorem for Cesaro limitability      38
Tauberian theorem for Cesaro summability      12
Tauberian theorem for circle methods, involving positivity condition      332
Tauberian theorem for circle methods, involving Schmidt condition      332
Tauberian theorem for general circle method, involving positivity condition      306
Tauberian theorem for general circle method, involving Schmidt condition      305
Tauberian theorem for general circle method, involving Valiron condition      303
Tauberian theorem for Ingham summability      110
Tauberian theorem for Lambert summability      18
Tauberian theorem for Lambert summability by Wiener theory      92 101
Tauberian theorem for method $J_{\mu}$       337
Tauberian theorem involving, (A,n) and (A,log n) summability      58
Tauberian theorem involving, Abel and (C,k) summability      17
Tauberian theorem involving, Abel and Cesaro summability      16 20 56
Tauberian theorem involving, Borel and Abel summability      60
Tauberian theorem involving, Dirichlet series      17 48—50
Tauberian theorem involving, Dirichlet series, extension      102
Tauberian theorem involving, family of kernels      90
Tauberian theorem involving, Gauss and Abel summability      61
Tauberian theorem involving, general series      100 102 103
Tauberian theorem involving, general-kernel transform      99
Tauberian theorem involving, harmonic function      113
Tauberian theorem involving, Lambert series      93 103
Tauberian theorem involving, Lambert transform      94
Tauberian theorem involving, Landau — Ingham asymptotics      107
Tauberian theorem involving, Laplace transform      28 30
Tauberian theorem involving, Laplace — Stieltjes transform      46
Tauberian theorem involving, Mellin — Stieltjes convolution      196
Tauberian theorem involving, power series      10 11 14—16
Tauberian theorem involving, power series, Schmidt form      95
Tauberian theorem involving, recurrent events      272
Tauberian theorem involving, series similar to power series      12
Tauberian theorem involving, Stieltjes series      101
Tauberian theorem involving, Stieltjes transform      47
Tauberian theorem involving, two-sided Laplace transform      233
Tauberian theorem of Fejer      12 276
Tauberian theorem of Milin      274
Tauberian theorem of nonstandard type      6 58 60 61

Tauberian theorem of Tenenbaum      281
Tauberian theorem of Zygmund      280
Tauberian theorem of Zygmund type      293
Tauberian theorem, classical form      6
Tauberian theorem, conditional converse of Abelian theorem      2
Tauberian theorem, evolution of concept      117
Tauberian theorem, high-indices theorem      18 50
Tauberian theorem, the name      2 15
Tauberian theorem, undoing averaging      2
Tauberian theorem, Wiener theorem      73 83
Tauberian theorem, Wiener — Pitt theorem      72 75 83 84
Tauberian theorem, Wiener's second theorem      96
Tauberian theorem, Wiener's second theorem for family of kernels      97
Taylor summability      329
Taylor summability, high-indices theorem      334
Taylor's formula, discrete analog      13
Tempered distribution      91
Tempered distribution, Fourier transform of      91
Tenenbaum, Abelian — Tauberian theorem for Borel summability      306
Tenenbaum, counting special prime factors      310
Testing equation      66
Testing equation for family of kernels      89
Testing equation for Wiener kernel      71 79 81
Testing equation for Wiener kernel, direct application      77 78
Testing equation for Wiener kernel, Fourier transformation      80 92
Testing equation, exponential solution      82
Testing function in distribution theory      90 155
Testing function, Schwartz space      91
Three-body problem and differentiation      37
Titchmarsh, Stieltjes type transforms      175 194
Transition from series to integral      25 67 347
Trapezoidal function      86 157
Trapezoidal function, trapezoidal testing function      92 157
Triangle function      80 86
Two-sided Laplace transform      232
Two-sided Laplace transform, Abelian theorem      233
Two-sided Laplace transform, Tauberian theorem      234
Typical mean      39
Unit step function, Heaviside function      156
Unit step function, polynomial approximation      50 352 359
Valiron, Stieltjes type transforms      175 194
Valiron, summability method      331
Valiron, Tauberian condition for Borel summability      302
Vallee Poussin (de la), PNT      19 63 118
Vanishing remainder      350 352 366 374
Very slowly decreasing function on $\mathbb{R}$       143
Very slowly decreasing partial sums      149
Very slowly oscillating partial sums      277
Very slowly varying function      185
Vijayaraghavan's method in boundedness theorem      40 280 296
Vitali's theorem      304
Vitali's theorem, applied to Tauberian theory      304
Weierstrass, Approximation Theorem      21 77
Weierstrass, kernel      399
Weight function      239
Weight function, condition of nonquasi-analyticity      241 246
Weight function, different conditions      250
Weight function, growth indices      240
Weighted $L^{1}$ space      239
Weighted $L^{1}$ space, dual space      240
Weighted-mean method of summability      341
Wielandt, variation on Karamata method      22
Wiener algebra      236
Wiener approximation theorem      83
Wiener approximation theorem, algebraic formulation      239
Wiener breakthrough      2 65 119
Wiener class M      95 216 307
Wiener condition on Fourier transform      70 81 82
Wiener condition on Mellin transform      196
Wiener family of kernels      89
Wiener family of kernels, characterization      89
Wiener family of kernels, special family      98 109
Wiener kernel      66 72
Wiener kernel, characterizations      82
Wiener kernel, Fourier transform      81
Wiener kernel, testing equation      70 72 79 82
Wiener kernel, translates      74 82
Wiener problem, admissible kernel      73 82
Wiener problem, comparing asymptotic averages      65 73
Wiener Tauberian conditions      95
Wiener Tauberian theorem      83
Wiener Tauberian theorem for series      100 103
Wiener Tauberian theorem, Beurling's extension      199
Wiener Tauberian theorem, comparing asymptotic averages      73 83
Wiener Tauberian theorem, involving family of kernels      90
Wiener theorem, algebraic form      239
Wiener theorem, approximation theorem      83
Wiener theorem, closed span of translates      83 89
Wiener theorem, distributional proof      92
Wiener theorem, division theorem      81 88
Wiener theorem, remainder theory for      398
Wiener theory      82
Wiener theory, background      65
Wiener theory, distributional approach      90
Wiener theory, extensions      85
Wiener — Ikehara theorem      67 119 122 124 127 162
Wiener — Ikehara theorem, finite form      128
Wiener — Ikehara theorem, finite form, sharp form      130
Wiener — Ikehara theorem, Graham — Vaaler method      119 128
Wiener — Ikehara theorem, open problem      134
Wiener — Ikehara theorem, refinement by Ingham      222
Wiener — Pitt problem      68 69
Wiener — Pitt problem, admissible kernel      69 70 72 82
Wiener — Pitt problem, testing equation      71
Wiener — Pitt theorem      66 72 75 83 84 379
Wiener — Pitt theorem, application      257 306 317
Wiener's lemma on Fourier series      66 84 243 273
Wiener's second Tauberian      95 96
Wiener's second Tauberian for family of kernels      97
Wiener's second Tauberian, application      214 216 218 307
Wiener's second Tauberian, extension involving Lebesgue — Stieltjes integral      98
Wiener's second Tauberian, special form      99
Wiener-type Tauberian theorem, involving positive kernel      76
Wonderful relation      64
Zeller      see "Functional-analytic method" "Meyer
Zero set for family of Fourier transforms      238
Zeta function      8 62
Zeta function, analytic continuation      62 94
Zeta function, analytic continuation, by Euler summability      325
Zeta function, critical strip      64 118
Zeta function, Euler product      8
Zeta function, in proof of prime number theorem      20 65 93 110 118 119 121 134 140
Zeta function, integral representation      62
Zeta function, Riemann hypothesis      64
Zeta function, zero-free line      63 95 122
Zeta function, zeros      64
Zorn's lemma      237
Zygmund ramp function      50
Zygmund special Tauberian theorem      280 292

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