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| Ïîèñê ïî óêàçàòåëÿì |
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| Korner T.W. — Fourier Analysis |
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| Ïðåäìåòíûé óêàçàòåëü |
Abel convergence test 539—540
Abel, reproached by Fourier 532
Adleman, secret codes 505 509
Airy, failures of imagination 30 333
Analytic function (and vice versa) 124 126—129 136—138
Analytic function Cauchy Riemann equations 121
Analytic function importance in number theory 546—556
Analytic function key theorems 379—380 382—383 397—399
Analytic function real part harmonic 122—123
Arnold, catastrophe theory 112
Author clad in a little brief authority 584
Author clad in nothing 395
Bachelier, economic Brownian motion 582—584
Bell, invention of telephone 25
Bessel function 174 486—487
Brown, discoverer of Brownian motion 43
Brownian motion, economic 581—584
Brownian motion, mathematical see “Separate entry”
Brownian motion, physical 43—45
Brownian path Levy's theorem 458
Brownian path properties 459
Brownian path, defined 458
Brownian path, used 459—460 467—469
Burgess Davis, proof of Picard's little theorem 467—469
Burt 429—441
Cambridge difficulty of exams, in old days 270—271
Cambridge mathematicians, corrupted by quantum mechanics 426
Cambridge mathematicians, Heaviside's sour view of 370
Cambridge probity of teachers 272
Carleson, convergence theorem 75
Casorati Weierstrass little theorem, proof via Brownian motion 459
Catastrophe theory 111—112
Cauchy distribution 246—252 583
Cauchy inversion formula, independent discovery of 295
Cauchy Liouville's theorem 175
Central limit theorem proof 347—356
Central limit theorem warnings 347 361
Central limit theorem, counter example 251—252
Central limit theorem, general form 357—361
Central limit theorem, multidimensional 413—417
Central limit theorem, used to construct Brownian motion 50
Character, defined 533
Character, principal 533
Characteristic function, another name for Fourier transform q.v. 245
Chebychev see “Tchebychev”
Chi squared distribution 415 423
Chi squared test 418—424 436—438
Clarke, A.M. and A.D.B., on Burt 435 440—441
Code making general criteria 503—505
Code making proposal of Rivest, Shamir and Adleman 509—511
Codebreaking, English Civil War 504—505
Codebreaking, World War II 92 503
Compass, Kelvin's work on 28—29 97
Complex variable see “Analytic functions”
Computers, early 30—31 47 92
Constructions by piling up of functions, divergent Fourier sum 67—73
Constructions by piling up of functions, nowhere differentiable function 38—41
Constructions by piling up of functions, sliding hump 73
Constructions by piling up of functions, unpleasant convolution 570—572
Constructions by piling up of functions, unpleasant Laplace transform 383—384
Convolution on Euclidean space natural occurrence 413—414 484—485
Convolution on Euclidean space, defined 412
Convolution on roots of unity, introduced 495—496
Convolution on roots of unity, neat trick, using 501—502
Convolution on the circle elementary properties 259—261
Convolution on the circle, defined 259
Convolution on the real line see also “Kernels”
Convolution on the real line counter example 570—572
Convolution on the real line in proof of Weierstrass's theorem 292—294
Convolution on the real line key slogan 262
Convolution on the real line properties 253—258 265
Convolution on the real line, better with Lebesgue integral 572
Convolution on the real line, connection with probability 253 258
Convolution on the real line, defined 253
Convolution on the real line, more general 565—572
Cooley, Fast Fourier Transform 499
Coprimality, probability of 528—531
Crystallography 481—483
Damped oscillator Duffing's equation 101—115
Damped oscillator stability 79—98
Damped oscillator, defined 79
Darwin golden rule 433
Darwin on Brown 43
Darwin Origin of Species 283 289—291
Data, how not to present 439
Data, how not to present (table) 430
Data, how to present 433
De Moivre's theorem see “Central limit theorem”
Demon of Chance, presides over financial markets 584
Differentiation and convolution 263 265—266 269
Differentiation under the integral sign 265—269
Dirichlet convergence of Fourier sums 3
Dirichlet encouraged by Fourier 480
Dirichlet L-function, defined 540
Dirichlet L-function, properties 537—538 541—550
Dirichlet problem see also “Laplace's equation”
Dirichlet problem for disc 121—123
Dirichlet problem uniqueness 124—125 135
Dirichlet problem with smoothness assumptions 124—130
Dirichlet problem without smoothness assumptions 131—141
Dirichlet problem, existence problem (partial solution) 125
Dirichlet problem, existence problem (possible non-existence) 139—140 140 163—164
Dirichlet problem, Monte Carlo method 47—49
Dirichlet theorem on primes in arithmetic progression 532—557
Dorfman, on Burt 435 438—439
Double periodicity, for analytic functions 175
Double tides 24
du Bois-Reymond, divergence of Fourier sums 3 67—73
Duffing's equation 101—115
Egyptology, Fourier's importance in 477
Eigen functions and values 181
Einstein special relativity 291
Einstein, Brownian motion 45
Ellipsoid, rotations of 416—417 420—421
Energy arguments for existence of solution of Dirichlet's problem 125
Energy arguments for existence of solution of Dirichlet's problem (and for uniqueness) 124—125
Energy arguments leading to Liapounov's method 79—81 84
Energy arguments, problems with 126 131—133 140 163—164
Equidistribution 11—14 47 408
Equiripple criterion, for best uniform approximation 202—206 212 215
Euclid algorithm 507
Euclid infinite number of primes 271 507 527
Euler Dirichlet formula 534—535
Euler generalisation of Fermat's little theorem 508
Euler remarkable formula 525—531
Euler remarkable formula (extended by Dirichlet) 532—535
Euler totient function 507—508
Euler, anticipates Fourier 170
Euler, infinite number of primes 527—528
Excessive optimism, Cauchy 3
Excessive optimism, Dedekind, Dirichlet and Weierstrass 67
Excessive optimism, Dirichlet, Gauss, Green, Kelvin and Riemann 126
Excessive optimism, Faraday and Morse 333
Excessive optimism, Galois 38
Excessive optimism, Hegel 370
Excessive optimism, Hermite and Poincare 42
Excessive optimism, Pearson 424
Excessive optimism, Poisson 119
Excessive optimism, Steiner 163
Excessive pessimism, Delambre 473—474
Excessive pessimism, general 4 74
Excessive pessimism, Lagrange 473
Excessive pessimism, Tchebychev 198
Eysenck, and Burt 429—432 434—435
Faraday 271 273 332—333
Fast factorisation no general method known 505 511
Fast factorisation work of Lenstra 512
Fast Fourier Transform 497—499
Fast multiplication 500—503
Fejer 4—5
| Fejer's theorem for Fourier transforms 223—225 240—243
Fejer's theorem proof and elementary consequences 6—10
Fejer's theorem, discussed 4—5
Fermat's little theorem 508
Fermi, Monte Carlo method 46
Fields, transatlantic cable 332—336
Fisher 424—425 427 439
Fourier analysis on groups, finite groups 513—518
Fourier analysis on groups, roots of unity 491—496
Fourier analysis on groups, techniques of book inadequate for non-Abelian groups 517—518
Fourier analysis on groups, techniques of book inadequate for non-Abelian groups (but work well for finite Abelian groups) 519—524
Fourier coefficients and convolution 259—261
Fourier coefficients Parseval's formula 156 183
Fourier coefficients uniqueness 9
Fourier coefficients, behaviour 153 260—261 409
Fourier coefficients, defined 3
Fourier discovers fundamental heat equation 25 478
Fourier generalises Fourier sums to non-trigonometric case 171—174
Fourier letter to Lagrange 170—171
Fourier life 475—480
Fourier linear programming 480 499
Fourier on behaviour of solutions of differential equations 88
Fourier sums see also “Sums”
Fourier sums best mean square approximation 148—149
Fourier sums Carleson's convergence theorem 75
Fourier sums Dirichlet's convergence theorem 3 61
Fourier sums divergence 3 67—75
Fourier sums near discontinuity 59—66 411
Fourier sums quality of approximation 28 35—37 151—152
Fourier sums rate of convergence 35—37 151—152
Fourier sums simplest convergence theorem 32—34 153 408 411
Fourier sums wider convergence theorem 57—58
Fourier sums, differentiation of 33 38
Fourier sums, multidimensional 409—411
Fourier sums, non-trigonometric (Sturm Liouville) 173—174 179—187
Fourier temperature of subsoil 25—27
Fourier transforms and convolution 257—258
Fourier transforms behaviour 263
Fourier transforms elementary properties 221—223
Fourier transforms heuristics 221
Fourier transforms in probability 245—252 347—356
Fourier transforms inversion theorem 295—298
Fourier transforms inversion theorem (with discontinuities) 300—307
Fourier transforms uniqueness 244
Fourier transforms, defined 221 580
Fourier transforms, multidimension 411—412
Fourier transforms, natural occurrence in optics 481—483
Fourier transforms, numerical calculation needed 481—487
Fourier transforms, numerical calculation of 488—490 497—499
Fourier views on mathematics 532 558—559
Fractals, not mentioned 112
Frobenius, ideas required for more advanced work 518
Gambling, advice on 442 450
Gauss Fast Fourier Transform 499
Gauss mean value theorem for harmonic functions 129 138
Gauss numerical integration 191—196
Gibbs phenomenon 62—66
Gibbs vectorial methods 370
Good sense, anglo saxon 291
Graduate students, use found for 92
Gram Schmidt orthogonalisation 188
Green, and Kelvin 271
groups see also “Fourier analysis on groups”
Groups of units modulo 506—508
Groups structure theorem for finite Abelian 519—524
Hadamard, shows failure of energy arguments 131—133
Haldane, life and opinions 425—428
Hardy keeps Titchmarsh in the dark 481
Hardy on English isolation 370
Hardy rejoices in remoteness of number theory 509—510
Hardy, Lebesgue measure essential for Fourier theory 572
Harmonic analyser 30—31 62
Harmonic function (function satisfying Laplace's equation) see “Under that entry”
Hausdorff, moment theorem 21—22 385
Heat equation and age of earth 285—288 370
Heat equation discovered by Fourier 25 478
Heat equation for cooling sphere 171—174
Heat equation for infinite rod 274—281
Heat equation for semi-infinite rod 287—288 308—323
Heat equation for temperature of subsoil 25—27
Heat equation for transatlantic cable 333—336
Heat equation solution by convolution 277—281
Heat equation solution by Fourier transforms 274—277
Heat equation uniqueness of solutions 277 344—346
Heat equation uniqueness of solutions (and non-uniqueness) 338—343
Heaviside, opinions 370—371 406
Helmholtz and Kelvin 336
Helmholtz on linearity 24—25
Herivel, biography of Fourier 170 475
Humboldt, patron of Dirichlet and Jacobi 556
Huxley, “Garbage in”, elegant formulation of 290
Images, method of 286—287
Inequalities for Fourier coefficients 153
Inequalities, Bessel 150
Inequalities, Cauchy 152
Inequalities, Cauchy, Schwartz, Buniakowski 146 152 180 492
Inequalities, isoperimetric see “Isoperimetric problem”
Inequalities, Tchebychev 47 251—252
Inequalities, triangle 147 181 492
Inner product and geometric intuition 147—148
Inner product associated with Sturm Liouville 180
Inner product on roots of unity 491—496
Inner product with weights 185
Inner product, discussed 145—158
Instability see “Stability”
Integration, numerical 46—47 191—196
Interchange of integrals easy for finite ranges 226—227
Interchange of integrals for infinite ranges 229 233—239
Interchange of integrals, counter examples 226 231
Interchange of sums, counter example 230—231
Interchange of sums, justified 231—233
Inversion theorems see “Fourier transform” “Laplace
Isoperimetric problem solution of Hurwitz via Fourier series 166—169
Isoperimetric problem solution of Steiner via geometry 159—163
Isoperimetric problem, discussed 159 164—165
Jacobi, annoyed by Poisson 532
Jacobi, enthused by Dirichlet 556
Kahane, divergence of Fourier sums 75
Kamin, on Burt 432 434 437
Katznelson, divergence of Fourier sums 75
Kelvin, Lord (Thomson), admires Liouville 174
Kelvin, Lord (Thomson), admires Peaucellier's linkage 200
Kelvin, Lord (Thomson), age of earth 283—291
Kelvin, Lord (Thomson), and Helmholtz 336
Kelvin, Lord (Thomson), founder of thermodynamics 271—272
Kelvin, Lord (Thomson), grandfather of computer 30—31
Kelvin, Lord (Thomson), intellectual relation to Faraday, Fourier, Green, Liouville and Maxwell 270—273
Kelvin, Lord (Thomson), invents mirror galvanometer 334
Kelvin, Lord (Thomson), leg pulled by Maxwell 575
Kelvin, Lord (Thomson), life 270—273 336—337
Kelvin, Lord (Thomson), method of images 286
Kelvin, Lord (Thomson), redesigns compass 28—29 97
Kelvin, Lord (Thomson), Thomson and Tait 273
Kelvin, Lord (Thomson), tidal prediction 29—31
Kelvin, Lord (Thomson), transatlantic cable 333—337
Kendall, economic Brownian motion 584
Kernels and convolution 254—255 261
Kernels as approximate identities 261
Kernels heat 279 292—293
Kernels on roots of unity 493—496
Kernels, Dirichlet 68—69
Kernels, Fejer 6—8 195 407
Kernels, modified Fejer 240—242
Kernels, Poisson 118—119
Kolmogorov axioms for probability 428
Kolmogorov divergent Fourier sums 74
Kolmogorov, Bachelier and Brownian motion 584
Kronecker's theorem, and equidistribution 408
L-function properties 537—538 541—550
L-function, defined 540
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