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| Korner T.W. — Exercises in Fourier Analysis |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel, behaviour on circle of convergence 67 100
Abel, brilliant stroke 155
Abel, convergence test 67
Abel, integral equation 277 281—283
Abel, Summation Formula 66 80
Abel, theorem on product of series 100
African Eve 371
Almost periodic functions 36—38 133
Apery, 'Miraculous demonstration' 172—173
Approximation see "Rate of convergence"
Bachelier, vindicated 375—377
Band limited functions 231—235 241—243
Bernoulli justifiable pride 69
Bernoulli numbers 68—71 78 222—223 263—264 see
Bernoulli polynomials 68—71 73—74 76
Bernstein approximation theorems 41—43 54
Bernstein inequalities 10 176 188—189 240—241
Bernstein polynomials 25—26 246
Bessel functions 321—325
beta function 273—275
Beta function, incomplete, Phillip Hall's view of 275
Beukers, proof of Apery's result 173—175
Black boxes 226—229 294—297
Blown up, Gibbs phenomenon 59
Bounded variation, functions of 55—58 204—205
Brent and Salamin, computation of pi 333—334
Brownian motion, construction of 366—369
Brownian motion, Fourier series of 372
Brownian motion, Levy's theorem on uniqueness of 374—377
Brownian motion, nowhere differentiable 371—372
Byrnes, hosts hostile tribes 242
Cantor, existence of transcendentals 144—145
Cantor, uniqueness theorem 111—113
Caratheodory's lemma for convex sets 185
CAT (computer assisted tomography) 318—319
Causality conditions 229 234—236 294
Central limit theorem 51
Champagne, good, offer of 139
Change of variable for Fourier transform 314
Change, how to count 259—262
Chebychev see "Tchebychev"
Compact discs (articles of commerce) 234
Concave function, Jensen's inequality for 124—125
Convergence of Fourier sums 55—58 200 203—205
Convex function see "Concave function"
Convex sets 122—123 185—186
Crinkles in nature 140 143
Cyber-punk Tripos question 371
de la Vallee — Poussin, prime number theorem 358—365
De Moivre's theorem 78—80
Depression, text book induced 151
Dido's problem 218
Difference equation, linear, how to solve 13—14
Differential equations, elementary functions, used to define 88—91
Differential equations, existence of solutions for 85—89
Differential equations, linear, how to solve 15 248—249
Differential equations, numerical solution of 93—94
Differential equations, rate of growth of solutions for 91—93
Dirichlet convergence test 66
Dirichlet evaluation of Gauss sum 329
Dirichlet lemma on approximation 34—35 299
Dirichlet original proof of convergence of Fourier sums 203—205
Dirichlet pigeonhole principle 35
Dirichlet problem, Neuman problem, related to 107—110
Dirichlet problem, obviously not soluble on physical grounds 116—117
Dirichlet problem, obviously soluble on physical grounds 115—116
Dirichlet problem, punctured plane, solution for 106
Dirichlet series 352 356—357
Divergent Fourier series, Fejer's example 80—81
Divergent Fourier series, Kahane and Katznelson, theorem of 82—84
Divergent Fourier series, Kolmogorov's everywhere divergent example 301—305
Divergent Fourier series, rate of divergence 64—66 202
Divergent Fourier transform 190
Doubly periodic functions see "Elliptic functions"
Eigenvalues via minimisation 157—163
Elliptic functions, properties and constructions 147—155 236—239
Elliptic functions, why so called 155
Elliptic integrals, equidistribution 11—13 15—16 130—133
Elliptic integrals, link with elliptic functions 155
Elliptic integrals, spectacular use of 332—334
Euler conjectures law of quadratic reciprocity 339
Euler equality (four squares) 300—301
Euler formula for Bernoulli numbers 70 222—223 263—264
Euler gamma 292 349 350 354
Euler infinite product for sine 285—286
Euler neat formula of 56
Euler superstar of superstrings 275
Euler Taylor series for tan 70
Euler totient function 335
Euler zeta of even integers 68 222—223
Euler — Maclaurin summation formula 61—62 71—77 131 350 353—354
Factorial function see "Gamma function"
Feedback see "Black boxes"
Fejer 11 80—81 181—183
Feller, proof of Stirling's formula 290—291
Feynman, anecdote 207
Fractional integral 275—277 281—283
Function see "Under specific name"
Gamma function, and Beta function 273—274
Gamma function, definition 270
Gamma function, duplication formula 271—273 288—290
Gamma function, factorial, natural extension of 271—273
Gamma function, formula involving 277 see "Watson's
Gamma, Euler's 292 349 350 354
Gauss computation of pi 284 331—334
Gauss formula for factorial function 272—273
Gauss proves law of quadratic reciprocity at age nineteen 338—339
Gauss sum 326—330 339
Generalised limit 7—8
Gibb's phenomenon 59—60 63—66
Green's function as transfer (or response) function 226—229 294—299
Green's function in complex variable theory 120
Green's function, found by firing pistol 228
Green's function, passing mention 163 225
Groechnig's algorithm for signal reconstruction 241—243
Guiness, does good by stealth 309
Hadamard gap theorem for natural boundaries 266—267
Hadamard prime number theorem 358—365
Hahn — Banach theorem in finite dimensions 123—124
Halmos, insufficiently asked question of 39
Hardy and Littlewood inequality 129
Hardy and Littlewood Tauberian theorem 102
Hardy nowhere differentiable function 46—47
Hardy test integral 60 208—209
Hardy transcendence of gamma, reward offered 349
Harpsichord 97—98
Harvard, mathematicians, mannerisms of 45
Hausdorff moment theorem 30—31
Hausdorff — Young inequality 136—139
Heisenberg inequality (uncertainty principle) 230—231
Herglotz theorem on positive definite sequences (simple version) 319—320
Hermite, quadrature formula 188
High definition TV 50
How not to index a book 381—385
How not to interpolate with polynomials 170 179—183
How not to throw dice 310
How not to write out a theorem 301 see
How to be almost sure a number is prime 342—343
How to build a harpsichord 97—98
How to conduct life 318
How to count change 259—262
How to expand in partial fractions 255—256
How to gamble, if you must 311—312
How to solve a linear difference equation 13—14
How to solve a linear differential equation 15 184—185
How to value stock options 377
Huygens, practical and theoretical work on pendulum clock 281—283
Inclusion-exclusion formula 348
Inequality, Bernstein (band limited functions) 240—241
Inequality, Bernstein (trigonometric polynomials) 10 176 188—189
| Inequality, Bessel 133
Inequality, Cauchy (arithmetic-geometric) 124
Inequality, Cauchy, Schwarz, Bumakowski 122
Inequality, Hardy and Littlewood 129
Inequality, Hausdorff — Young 136—139
Inequality, Heisenberg 230—231
Inequality, Hilbert 128—129
Inequality, Hoelder 125
Inequality, Hoelder, converse 126
Inequality, Jensen (concave functions) 124—125
Inequality, Jensen (zeros of analytic functions) 114—115 288
Inequality, Tchebychev (polynomials) 188—189
Inequality, Tchebychev (probability) 25 79 197—198
Inequality, triangle 122 125 240
Inequality, Van der Corput (first and second) 129—131
Inequality, Van der Corput (related to equidistribution) 130—131
Inequality, Wirtinger 140 162 239—240
Infinite products 77 284—291
Insects, love life of 214 302
Interpolation by polynomials 25—27 170 178—183
Interpolation formula of Rivlin and Shapiro 185—189
Jackson approximation theorems 41 43—45 54
Jacobi, compulation of Legendre symbol 340—342
Jacobi, identity 238—239 354—356
Jacobi, remainder in Euler — Maclaurin formula 73
Jacobi, symbol 340—342
Jacobi, theta function 236—239
Jigsaw puzzle, mathematician's 231
Kahane and Katznelson, divergent Fourier series 82—84
Kahane, beautiful probabilistic arguments 261 371—372
Karamata, proof of Tauberian theorem 102—103
Katznelson see "Kahane and Katznelson"
Kolmogorov everywhere divergent Fourier series 301—305
Kolmogorov zero-one law 267—9
Kramers — Kronig (causality) relations 234—236
Kronecker's theorem (on simultaneous approximation), another proof 337
Kronecker's theorem (on simultaneous approximation), used 301—305
Lacunary series 266—270
Lagrange's four squares theorem 300—301
Landau driven out of Gottingen 91
Landau sage advice 41
Laplace transforms, convolution, for 257—258
Laplace transforms, delicate treatment of 363—364
Laplace transforms, partial differential equations, used with 258
Laplace transforms, warning 93 see
Lattes, remarkable rational function 153—154
Lebesgue, proof of Weierstrass theorem 27—28
Lebesgue, thorn 115—117
Legendre symbol 338—343 345—346
Lerch's theorem, by complex variable 287 288
Levy's theorem on uniqueness of Brownian motion 374—376
Lipschitz condition 87—88
Littlewood see also "Hardy and Littlewood"
Littlewood, obiter dicta 138 301
Matrices, fun with 95—96 157—158 248—251 301 307—308 314 327—328
Mean value theorems for integrals 67 202—203
Mellin transform 343—344 351—352
Minkowski, inequality 125
Minkowski, theorem on geometry of numbers 299—301
Monte Carlo method 51—52
Mordell, neat evaluation 198
Mordell, proof of Hadamard gap theorem 266—267
Mordell, suitable thesis topic 349
Natural boundary for analytical functions 264—270 286
Neumann problem 107—110
Newman D.J., makes hard things less hard 333—334 363—364
Newton — Cotes formula, problems with 169—70 278—279
Newton, sums of powers of roots 15—16
Nowhere differentiate functions, Brownian motion 371—372
Nowhere differentiate functions, Hardy 46—47
Nowhere differentiate functions, Van der Waerden 48
Nowhere differentiate functions, Weierstrass 46—47
Numerical analysts, wise sayings of 43
Numerical quadrature (integration) 168—170 180 278—279
Nyquist, criterion for feedback 296—297
Nyquist, rate for signal transmission 231—232 241—242
Oven, microwave, unsatisfactory 33
Pal, interpolation by polynomials with integer coefficients 26—27
Paley — Wiener type theorem 234
Parseval's equality for Fourier transforms 224 240—241
Partial fractions, how to get 255—256
Partial fractions, not as boring as they look 260—261
Peano, kernel theorem 21—23
Peano, space filling curve 49—50
Pendulum clock 281—283
Periodicities, hidden 34
Pi, computation of 77 284 331—334
Pisot number 15—16
Poisson, formula 61 222 231 239
Poisson, summation 100 103—105
Polya, infinity of primes 211
Polya, numerical quadrature 169—170 278—279
Polya, random walks 312—313
Polynomial, Bernoulli 68—71 73—74 76
Polynomial, Bernstein 25—26 246
Polynomial, Laguerre 166—167
Polynomial, Legendre 164—166 175
Polynomial, Rudin — Shapiro 138—139
Polynomial, Tchebychev 25 105 168—169 188—189
Polynomials, interpolation by 25—27 170 178—183
Polynomials, sums of powers of roots 15—16
Population, explosion 370
Population, oscillation 251—253
Positive definite quadratic forms 95—96
Positive definite quadratic sequencies 319—320
Potatoes, two, and string, what to do with 98—99
Prime number theorem, full, of Hadamard and de la Vallee — Poussin, proved 358—365
Prime number theorem, full, of Hadamard and de la Vallee — Poussin, used 173—175
Prime number theorem, limited, of Tchebychev, proved 171—173
Prime number theorem, limited, of Tchebychev, used 343 362—363
Prime numbers, probabilistic test for 342—343
Primes, infinity of Euclid 211
Primes, infinity of Polya 211
Radar, electronically steered 315—317
Radon transform 318—319
Random series, construction of Brownian motion using 366—369
Random series, example given by Fourier series of Brownian motion 372—374
Random series, Taylor, and natural boundaries 269—270
Random walk returns 312—313
Rate of convergence of approximations 41—45 53—54
Reconstruction of band limited signals 231—234 241—243
Response function (transfer function) 226—229 294—297
Richardson L.F., deferred approach to the limit 73 75
Riemann integral, unsatisfactory 127—128 135
Riemann — Lebesgue lemma 10 200—201 220
Riemann, Fourier series, work on 111—113
Riemann, fractional integrals 275—279
Riemann, functional equation for zeta 354—356
Riemann, hypothesis 361 364—365
Riemann, localisation (weak version) 201
Riemann, Mapping Theorem 118—121
Riemann, thesis report on 121
Riemann, zeta function see "Main entry"
Rivlin and Shapiro interpolation formula 185—189
Salamin see "Brent and Salamin"
Schur, I., evaluation of Gauss sum 326
Schur, I., proof of two inequalities of Hilbert 128—129
Schur, I., theorem on making change 259—261
Set theory, origin of 113
Shannon sampling theorem 231—234
Shapiro see also "Rivlin and Shapiro"
Shapiro, MSc thesis 138
Simpsin's rule, error bounds for 22—23
Smith, genealogy applied to Brownian motion 371—372
Smith, why are we all called 371
Smith, why are we not all called 369—371
Smoke rings 216
Solvay and Strassen's probabilistic test for primes 342—343
Space filling curve of Liu Wen 49—50
Sphere, heating and cooling 32—33 216—218
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