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Widder D.V. — The Laplace transform
Widder D.V. — The Laplace transform

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Название: The Laplace transform

Автор: Widder D.V.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1941

Количество страниц: 406

Добавлена в каталог: 06.12.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$d_{t}\beta(t,\tau)$      50
$\sigma_{a}$      46 47
$\sigma_{a}$ - $\sigma_{c}$ for Dirichlet series      49
$\sigma_{a}$ - $\sigma_{c}$ for the Laplace transform      47
$\sigma_{c}$      37 42 43
$\sigma_{u}$      51 52 53
(C,p) equivalent to (H,p)      123
(H,p), (C,p) equivalent to      123
Abel      89 116
Abelian theorems      181 185
Abelian theorems for the Laplace transform      180
Abelian theorems for the Stieltjes transform      183
Abscissa of convergence of the bilateral Laplace transform      238
Abscissa of convergence of the bilateral Laplace transform, Dirichlet series      45
Abscissa of convergence of the bilateral Laplace transform, Laplace transform      37 38 42
Abscissa of convergence of the uniform convergence for the Laplace transform      51
Abscissas of convergence of the bilateral Laplace transform      240
Abscissas of convergence of the bilateral Laplace transform, absolute convergence of the bilateral Laplace transform      241
Absolute convergence of the Laplace transform      46
Absolutely convergent improper Stieltjes integrals      15
Absolutely convergent improper Stieltjes integrals, monotonic ($\Delta$)      148
Absolutely convergent improper Stieltjes integrals, monotonic (D)      147
Absolutely convergent improper Stieltjes integrals, monotonic (G)      154
Absolutely convergent improper Stieltjes integrals, monotonic function      144
Absolutely convergent improper Stieltjes integrals, monotonic functions, additional properties of      156
Absolutely convergent improper Stieltjes integrals, monotonic functions, analytic extension of      146
Absolutely convergent improper Stieltjes integrals, monotonic functions, analyticity of      146
Absolutely convergent improper Stieltjes integrals, monotonic functions, derivatives of      167
Absolutely convergent improper Stieltjes integrals, monotonic functions, equivalence of Bernstein and Gruess definitions      155
Absolutely convergent improper Stieltjes integrals, monotonic functions, equivalence of Bernstein and Gruess definitions, Bernstein's two definitions      151
Absolutely convergent improper Stieltjes integrals, monotonic functions, existence of one-sided derivatives of      149
Absolutely convergent improper Stieltjes integrals, monotonic functions, Gruess's definition of      154
Absolutely convergent improper Stieltjes integrals, monotonic functions, higher differences of      150
Absolutely convergent improper Stieltjes integrals, monotonic functions, series of      151
Absolutely convergent improper Stieltjes integrals, monotonic functions, with prescribed derivatives      165
Additional properties of absolutely monotonic functions      156
Analytic character of the generating function      57
Analytic character of the Stieltjes transform      328
Analytic extension of absolutely monotonic functions      146
Analyticity of absolutely monotonic functions      146
Application of Wiener's theorem      221 222
Applications of the Laplace method      280
Arithmetic means      76
Asymptotic evaluation of an integral, Laplace's      277 296
Asymptotic evaluation of an integral, Laplace's, properties of Stieltjes transforms      329
Axis of convergence of the bilateral Laplace transform      238
Axis of convergence of the Laplace transform      37
Axis of uniform convergence for the Laplace transform      50
Banach      33
Bernoulli numbers      264
Bernstein      144 147 165 168
Bernstein polynomials      101 152
Bernstein's theorem      160 162 175 312
Bessel's function      169
Bieberbach      95
Bilateral Laplace integrals, product of      257
Bilateral Laplace transform      237
Bilateral Laplace transform, abscissa of convergence of      238
Bilateral Laplace transform, abscissas of, absolute convergence of      241
Bilateral Laplace transform, abscissas of, convergence of      240
Bilateral Laplace transform, axis of convergence of      238
Bilateral Laplace transform, integration by parts of      239
Bilateral Laplace transform, necessary and sufficient conditions for representation as      272
Bilateral Laplace transform, region of convergence of      238
Bilateral Laplace transform, summability of      244
Bilateral Laplace transforms, product of      252
Bilateral Laplace transforms, representation of functions as      265
Bilateral Laplace — Lebesgue transform, inversion formulas for      241
Bilateral Laplace — Lebesgue transforms      258
Bilateral Laplace — Stieltjes transform, inversion formulas for      242
Bilateral Laplace — Stieltjes transform, Stieltjes resultant      248 249 250
Blumenthal      135
Boas      18 138 177 275 363 383 384 386
Bochner      252 273
Bohr      53
Bonnet's second law of the mean      17
Borel      337
Bound, greatest lower      4
Bound, least upper      4
Bounded functions      111
Bounded functions, variation, determining function of      306
Bounded functions, variation, function $\alpha(t)$ of      361
Bounded functions, variation, functions of      6
Bounded functions, variation, normalization of functions of      13
Bounded functions, variation, sequence of functions of uniformly      31
Bray      25 31
Cahen      246
Carleman      177
Carlson      161
Cauchy's integral theorem      265
Cauchy's singular integral      338
Cauchy's theorem      86 88
Cauchy-value      15 35
Cesaro's means      76
Cesaro's method of order p, (C,p)      121
Cesaro's summability      114 118 119
Change of variable of a Stieltjes integral      19
Class $L^{p}$      312 368
Class A      204
Class A, norm of a function of      205
Class B      209
Class L      317 374
Class M      213
Class S      209
Class S*      211
Class W      210
Classical resultant      91
Completely convex functions      177
Completely monotonic functions      144 161 310
Completely monotonic functions, interpolation by      163
Completely monotonic sequence      158 269
Completely monotonic sequence, minimal      163
Completely monotonic sequences      108
Complex inversion formula for the Laplace transform      63
Complex inversion formula for the Stieltjes transform      338
Complex variable Tauberian theorem      233
Condition A      101
Condition A for a moment sequence      101
Condition B for a moment sequence      109
Condition C for a moment sequence      111
Conditions A for the Laplace transform      306
Conditions A for the Stieltjes transform      361
Conditions A' for the Laplace transform      308
Conditions B for the Laplace transform      312
Conditions B for the Stieltjes transform      369
Conditions C for the Laplace transform      315
Conditions C for the Stieltjes transform      372
Conditions D for the Laplace transform      317
Conditions D for the Stieltjes transform      374
Conditions of integrability for the Stieltjes integral      24
Consistent      115
Consistent, methods of summability      115
Convergence properties of the Laplace integral      38
Convergent improper Stieltjes integrals      15
Convex functions      148 167
Convex functions, completely      177
Convexity      148
Convolution      84
de la Vallee-Poussin      224
Definite quadratic forms      136
Derivatives of absolutely monotonic functions      167
Determinant, Hadamard's maximum value of a      141
Determinant, Hadamard's maximum value of a, criteria for definite sequences      134
Determinants, Hankel      135
Determining function      38
Determining function of bounded variation      306
Determining function, belonging to $L^{2}$      80 245
Determining function, fractional integral of      73
Determining function, integrals of the      70
Determining function, non-decreasing      310
Determining function, normalized      237
Determining function, periodic      96
Determining function, the integral of a bounded function      315
Determining function, uniqueness of the      59
Determining function, variation of the      299
Diagonal matrix      114
Difference matrix      114
Dini's condition      67
Dirichlet integral      64
Dirichlet series      44 53
Dirichlet series, $\sigma_{a}$ - $\sigma_{c}$ for      49
Dirichlet series, abscissa of convergence of      45
Divergent improper Stieltjes integrals      15
Divergent integrals, summability of      75
Doetsch      168
DUBOURDIEU      295
Elementary properties of the Stieltjes transform      325
Entire function, order of an      94 179
Entire function, type of an      94 179
Entire function, type of an, functions      177
Entire function, type of an, generating function      58
Equivalence of Bernstein and Gruess definitions of absolutely monotonic functions      155
Equivalence of Bernstein's two definitions of absolutely monotonic functions      151
Equivalent methods of summability      121
Euler's constant      229
Euler's Constant, theorem for homogeneous functions      291 303 357 385
Evans      31
Existence of one-sided derivatives of absolutely monotonic functions      149
Existence of Stieltjes integrals      7
Extension of moment operator      127
Faber      96
Factorial series      97
Faltung      64
Fatou's lemma      314
Feller      295
Fort      99
Fourier development      96
Fourier integrals      204
Fourier transform, uniqueness of      82 204
Fourier transforms      202
Fourier transforms, of functions of L      204
Fourier transforms, product of      203
Fourier transforms, quotient of      207
Fourier — Stieltjes transforms, product of      252
Fractional integral      71
Fractional integral, of determining function      73
Fredholm, function of      157
Fubini's theorem      26 41 71 73 74 82 200 201 203 207 211 256 259 266 292 318
Function of $L^{p}$      109
Function, $\alpha(t)$ non-decreasing and bounded      363
Function, $\alpha(t)$ non-decreasing and unbounded      365
Function, $\alpha(t)$ of bounded variation      361 377
Function, $\varphi(t)$ bounded      372
Function, absolutely monotonic      144
Function, completely monotonic      144 310
Function, normalized determining      237
Function, order of an entire      179
Function, type of an entire      179
Functional, linear      171
Functions analytic at infinity      93
Functions of $L^{p}$      11
Functions of bounded variation      6
Functions of bounded variation, moments of      138
Functions of bounded variation, normalization of      13
Functions of L, Fourier transforms of      204
Functions, bounded      111
Functions, completely convex      177
Functions, completely convex, monotonic      161
Functions, convex      148 167
Functions, entire      177
Functions, logarithmically convex      167
Functions, number theoretic      224
Functions, Riemann-integrable      190
Functions, slowly decreasing      209
Functions, slowly oscillating      209
Gauss      224
General Laplace — Stieltjes transform      320
General Laplace — Stieltjes transform, representation theorem for the Laplace kernel      302
General Laplace — Stieltjes transform, representation theorem for the Stieltjes kernel      355
General Laplace — Stieltjes transform, theorems for the Stieltjes transform      360
Generating function      37
Generating function as factorial series      97
Generating function, analytic at infinity      93
Generating function, analytic character of the      57
Generating function, entire      58
Generating function, singularities of      58
Goursat      337
Greatest lower bound      4
GRUESS      154
Gruess's definition of absolutely monotonic functions      154
hadamard      224
Hadamard's maximum value of a determinant      141
Half-plane of absolute convergence for the Laplace transform      46
Hamburger      58 125 265 266
Hamburger moment problem      125 129
Hamburger moment problem, not unique      125
hankel      135
Hankel determinants      135 167
Hardy      71 77 78 161 169 193 198 199 370 387
Hausdorff      100 113 144 163
Hausdorff matrix      118 119
Hausdorff moment problem      100 273
Hausdorff summability      113 115
Hausdorff transformation      118
Helly      26 31
Helly — Bray theorem      31 33 253
Helly's theorem      104 176 307 367
Higher differences of absolutely monotonic functions      150
Hilbert, inequality of      370
Hildebrandt      3 105
Hille      113
Hobson      76
Hoelder summability      114 118 119
Hoelder's inequality      110 313 314 370
Hoelder's method of summability of order p, (H,p)      120
Homogeneous functions, Euler's theorem on      291 303 357 385
Homogeneous Laplace integral equation      390
HYSLOP      387
Ikehara's theorem      233
Improper Stieltjes integrals      15
Indefinite integrals, Stieltjes resultant of      256
Indeterminacy of solution of moment problem      142
Inequalities for Stieltjes integrals      9
Integral equation      390
Integral equation, of a bounded function, the determining function      315
Integral equation, solution of an      387
Integral equation, with Laplace kernel      390
Integral equation, with Stieltjes kernel      387
Integrals of the determining function      70
Integration by parts for the Stieltjes integral      8
Integration by parts of the bilateral Laplace transform      239
Integration by parts of the Laplace integral      41
Integration with respect to an indefinite Stieltjes integral      12
Interpolation by completely monotonic functions      163
Inversion formula for the Laplace transform      386
Inversion formula for the Laplace — Stieltjes integral      69
Inversion formulas for the bilateral Laplace — Lebesgue transform      241
Inversion formulas for the Laplace — Stieltjes transform      242
Inversion of the Laplace transform      276
Inversion operator for the Laplace — Lebesgue transform      288
Inversion operator for the Laplace — Stieltjes transform      290
Inversion operator for the moment sequences      107
Inversion operator for the Stieltjes transform      345 347
Inversion when the determining function belongs to $L^{2}$      80
Iterated Stieltjes integrals      25
Iterated transform      383
Iterates of the Stieltjes kernel      259 262
Iteration of the Laplace transform      325
Jump operator for the Laplace transform      298
Jump operator for the Stieltjes transform      351
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