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.

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

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Предметный указатель

Abel sum      1 4
Abel summability      1 4 25 67 268
Abel summability of Fourier series      7
Abel summability of integral      26 61
Abel summability, comparison with Borel summability      60
Abel summability, gap Tauberian theorem      2 269 see
Abel summability, Tauber's theorems for      10
Abel summability, Tauberian theorem      14 15
Abel summability, Tauberian theorem, remainder in      17 348
Abel summability, Wiener form      380
Abel transform      117 268
Abel's continuity theorem      1 4 117
Abelian theorem      2
Abelian theorem for Borel summability      5
Abelian theorem for Cesaro summability      3
Abelian theorem for Cesaro to Abel summability      4
Abelian theorem for Euler to Borel summability      328
Abelian theorem for Lambert summability      5
Abelian theorem in Wiener — Pitt problem      68
Abelian theorem involving, Mellin transform (complex)      123
Abelian theorem involving, Mellin — Stieltjes convolution      195
Abelian theorem involving, power series      16
Abelian theorem involving, two-sided Laplace transform      233
Abelian theorem, continuity theorem      2
Abelian theorem, the name      5
Abelian — Tauberian combination involving, general circle method      306
Abelian — Tauberian combination involving, Laplace transform      193 194 197
Abelian — Tauberian combination involving, Stieltjes transform      194 196
Abelian — Tauberian combination involving, two-sided Laplace transform      234
Acceleration of convergence      325
Admissible kernel in Wiener problem      73 82
Admissible kernel in Wiener — Pitt problem      69 70 72 82
Admissible kernel testing equation      70
Algebraic form of Wiener theorem      237
Amalgam norm      195
Analytic continuation      288 see
Analytic continuation by Borel summability      7 279 284—286
Analytic continuation by Euler summability      325
Analytic continuation of zeta function      62 94 325
Analytic continuation, Borel — Okada principle      288
Analytic continuation, Taylor method      329
Application of Tauberian theory to      see also "Next item"
Application of Tauberian theory to Fourier series      13
Application of Tauberian theory to operator theory      163
Application of Tauberian theory to partitions      228
Application of Tauberian theory to prime number theory      134 310 402
Application of Wiener theory to Borel Tauberian      305
Application of Wiener theory to general circle method      305 307
Application of Wiener theory to Hardy — Littlewood theorem      77 95 101
Application of Wiener theory to harmonic functions      111 114
Application of Wiener theory to Ingham summability      110
Application of Wiener theory to Lambert summability      92 101
Application of Wiener theory to Landau — Ingham asymptotics      107
Application of Wiener theory to Littlewood's theorem      77
Application of Wiener theory to prime number theorem      93 110 111
Application of Wiener theory to renewal theory      272
Application of Wiener theory to strong asymptotics      214 216 218
Application of Wiener theory to Tauberian theorem for power series      103
Approximate identity      80 125 239
Approximation theorem of Beurling      241
Approximation theorem of Weierstrass      21 77
Approximation theorem of Wiener      83
Approximation theorem, Muentz — Szasz formula      318
Approximation theorem, one-sided approximation      352
Area condition of conformal mapping      12
Arzela theorem      71
Asymptotic averages, comparison      65 72 77 78 82
Asymptotics      see also "Abelian..." "Tauberian..."
Asymptotics for derivatives      34 35
Asymptotics, involving counting      194 225 228 310
Asymptotics, involving Laplace transform      192 197
Asymptotics, involving partitions      228
Asymptotics, involving Stieltjes transform      194
Asymptotics, Landau — Ingham      107
Asymptotics, large Laplace transforms      207
Asymptotics, large Laplace transforms, boundedness result      209
Asymptotics, large Laplace transforms, complex theory      222
Asymptotics, large Laplace transforms, logarithmic theory      208 219
Asymptotics, large Laplace transforms, strong      208 212 214 218
Bad approximability and high-indices theorems      54
Banach algebra      85 236
Banach algebra, $A_{\omega}=(L_{\omega},\mathbb{C})$       244
Banach algebra, $A_{\omega}=(L_{\omega},\mathbb{C})$ , maximal ideal      247
Banach algebra, $L^{1}$ under convolution      85 236
Banach algebra, adjoining unit element      244
Banach algebra, Beurling algebra      240
Banach algebra, coset algebra      241
Banach algebra, Gelfand representation      243
Banach algebra, Gelfand theory      241
Banach algebra, Gelfand transform      243
Banach algebra, homomorphism      242
Banach algebra, ideal      237
Banach algebra, invertible element      237 242
Banach algebra, maximal ideal space      243
Banach algebra, normed field      241
Banach algebra, regular algebra      246
Banach algebra, unit element      236
Banach algebra, weighted $L^{1}$ space      240
Banach algebra, Wiener algebra      236
Banach coordinate space, B K-space      259 262
Banach space      85
Berry — Esseen inequality      232 384
Beurling algebra      see "Convolution algebra $L_{\omega}$ "
Beurling approximation theorem      241 249
Beurling approximation theorem, open problem      241
Beurling slow variation      184 191 199
Beurling slow variation, open problem      200
Beurling slow variation, self-neglecting function      200
Beurling slow variation, Tauberian theorem      199
Bieberbach — de Branges inequality      274
Big $\mathcal{O}$ -condition      see "Tauberian condition"
Bingham et al., average-type Tauberian conditions      16 58 321 333
Bingham et al., book Regular Variation      178
Bingham et al., circle methods      281 309 333
Blaschke product      375 377
Borel analytic continuation      279
Borel density      310
Borel function      288 310
Borel polygon      285
Borel sum      5
Borel summability      2 5 279 282
Borel summability of power series      283 285
Borel summability, analytic continuation      7 284—286
Borel summability, boundedness theorem      296
Borel summability, comparison with (C,k) summability      333
Borel summability, comparison with Abel summability      60
Borel summability, Hardy — Littlewood theorem      18 302
Borel summability, high-indices theorem      280 312
Borel summability, Schmidt condition      279
Borel summability, Tauberian condition      279
Borel summability, Tenenbaum theorem      306
Bounded analytic function, Jensen theorem      377
Bounded analytic function, set of smallness      366
Boundedness theorem involving, Borel summability      296
Boundedness theorem involving, FK-space      257 260 262
Boundedness theorem involving, gap series      50 52
Boundedness theorem involving, general circle method      295
Boundedness theorem involving, general-kernel transform      43
Boundedness theorem involving, Laplace transform      143
Boundedness theorem involving, method $J_{\mu}$       337
Boundedness theorem involving, monotone minorant      40 296
Boundedness theorem involving, power series      11 40
Boundedness theorem involving, power series, related series      12
Boundedness theorem involving, rapidly decreasing kernel      235 250 316
Calculus of finite differences      325
Calculus of finite differences, symbolic operators      325
Cauchy formula      136 226 273 286 405
Cauchy functional equation      185
Cauchy product, Cesaro summability      6
Cesaro (C,k) mean      39

Cesaro (C,k) mean of integral      37
Cesaro sum      1 3
Cesaro summability      1 3 16 25 37 67 260 261 267
Cesaro summability of Cauchy product      6
Cesaro summability of Fourier series      6
Cesaro summability of gap series      14
Cesaro summability of integral      26 79
Cesaro summability of order k      17
Cesaro summability of order k of integral      38
Cesaro summability of order k of series      39
Cesaro summability, gap Tauberian theorem      269
Cesaro summability, remainder theorem for      14
Cesaro summability, Tauberian theorem      12
Characteristic function of set      182
Characteristic function, Fourier transform of distribution function      232
Chebyshev function      19 107 122 140
Chebyshev polynomial      358
Christoffel numbers      361
Circle methods of summability      280 328
Circle methods of summability as special cases of general circle method      331 332
Classical Dirichlet series      see "Dirichlet series"
Closed (linear) span      259
Closed (linear) span of translates in $L^{1}$       73 83 89
Closed (linear) span of translates in other spaces      83
Complex Fourier transform      240 379
Complex remainder theory      119 165
Complex remainder theory for Fatou theorem      167
Complex remainder theory for Stieltjes transform      173
Complex remainder theory, auxiliary results      165 166
Complex remainder theory, involving power series      170 172
Complex Tauberian      118
Complex Tauberian in operator theory      163—165
Complex Tauberian involving, Dirichlet series      120 122 133 138 149
Complex Tauberian involving, Laplace transform      124 135 141 142 154 161 222
Complex Tauberian involving, Mellin transform      124
Complex Tauberian, (Laplace) finite form      138
Complex Tauberian, Fatou theorem      118 148
Complex Tauberian, Fatou theorem, extensions      149—151 153 157 159
Complex Tauberian, Ingham theorem      133
Complex Tauberian, Landau theorem      120
Complex Tauberian, strong Ingham theorem      222
Complex Tauberian, Wiener — Ikehara theorem      119 122 124
Concentration function      232
Conformal mapping      367 376
Conformal mapping, area condition      12
Conformal mapping, Bieberbach — de Branges inequality      274
Conformal mapping, Hayman regularity theorem      274
Conjugate convex function      233
Conservative summability method      258
Consistent methods      3
Continuity theorem      see "Abelian theorem"
Continuity Theorem (the), for Laplace — Stieltjes transform      192
Continuous linear functional on $C^{\infty}_{2\pi}$       155
Continuous linear functional on $L^{1}$       74
Continuous linear functional on $L_{\omega}$       240 249
Continuous linear functional on FK-space      259
Continuous linear functional on Schwartz space      91
Contour integration method of Newman      119 133
Contour integration method of Newman, extension      135 136 151
Convexity theorem, involving derivatives      37
Convexity theorem, involving sequences      14
Convolution      80
Convolution algebra $L^{1}$       236 237
Convolution algebra $L^{1}$ , form of closed maximal ideal      237 239
Convolution algebra $L^{1}$ , minimal ideal at $\infty$       238
Convolution algebra $L^{1}$ , translation invariant subspace      238
Convolution algebra $L^{1}$ , Wiener approximation theorem      239
Convolution algebra $L_{\omega}$       240
Convolution algebra $L_{\omega}$ , Beurling theorem      241
Convolution algebra $L_{\omega}$ , closed maximal ideal      248 250
Convolution algebra $L_{\omega}$ , different classes of $\omega$       250
Convolution algebra $L_{\omega}$ , regularity condition      246
Convolution of sequences      400
Convolution of sequences, Erdos problem      400 401
Convolution of sequences, Erdos theorem, application to prime number theory      402
Convolution of sequences, open problems      409 412
Convolution of sequences, square-root problem      401
Convolution of sequences, third problem      402
Convolution, Fourier transform      80
Convolution, Mellin — Stieltjes      195
Counting (special) prime factors, Halasz relation      310
Counting (special) prime factors, Mangoldt (von) theorem      310
Counting (special) prime factors, Tenenbaum theorem      310
Counting function      177
Counting function for eigenvalues      175
Counting function for zeros      175
Critical rate, in remainder theory      350 393
Critical strip, zeta function      64
Darboux property of derivative      36
de Haan theory      189 190 200
de Haan theory, class $\Pi$       189
de Haan theory, class $\Pi^{-}$       189 221
de Haan theory, class $\Pi^{-}$ , characterization      221
Differences and derivatives      35
Differentiation of asymptotic relation      34 35 37 121
Dirichlet series, classical Dirichlet series      8 17
Dirichlet series, complex Tauberian      120 122 133 138 149
Dirichlet series, Fatou — Riesz theorem      118 149
Dirichlet series, gap theorem of Ingham      150
Dirichlet series, high-indices theorem      50
Dirichlet series, remainder estimate      373
Dirichlet series, representation by      8
Dirichlet series, Tauberian theorem      17 48 49
Dirichlet series, Wiener — Ikehara theorem      122
Dirichlet series, zeta function      8 62
Discrete Taylor formula      13
Distribution function      232
Distribution function, estimate using Fourier transform      232
Distribution function, estimate using Laplace transform      232
Distribution theory (generalized functions)      90
Distribution theory (generalized functions), Fourier transformation      91
Distribution theory (generalized functions), periodic distribution      155
Distribution theory (generalized functions), periodic testing function      155
Distribution theory (generalized functions), Schwartz space      91
Distribution theory (generalized functions), tempered distribution      91
Distribution theory (generalized functions), testing function      91
Distributional convergence      155 161
Distributional proof of Wiener theorem      92
Divergent series involving zeta function      59 135
Division theorem of Wiener      81 88
Division theorem of Wiener, proof      88
Division theorem of Wiener, refinement      251
Division theorem of Wiener, special cases      86 87
Divisor function      9
Drasin — Shea theorem      204
Effective domain      258
Effective domain, gap-perfect part      267
Effective domain, seminorms      258
End game      417
Equivalent summability methods      4
Erdos, (and Feller and Pollard) renewal theory      271 272
Erdos, (and Selberg) PNT      63 118
Erdos, convolution problem      400
Erdos, Erdos — Hua problem      403
Erdos, nonlinear Tauberian      402
Erdos, nonlinear Tauberian, open problem      412
Euler constant      9 62 107 140 200
Euler product      8
Euler summability      260 325 326
Euler summability to Borel summability      328
Euler summability, gap Tauberian theorem      270
Euler summability, methods (E,q), $E_{\alpha}$       328
Euler transformation      325 326
Exponentiation of power series      274
Extremal kernel      146
Extremal majorant of type $\lambda$       129
Fabry gaps      323
Fabry theorem      294 335
Fatou theorem for harmonic function      112
Fatou theorem for harmonic function, analog      113

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