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Korner T.W. Ч Fourier Analysis
Korner T.W. Ч Fourier Analysis

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Ќазвание: Fourier Analysis

јвтор: Korner T.W.


Ranging from number theory, numerical analysis, control theory and statistics, to earth science, astronomy and electrical engineering, the techniques and results of Fourier analysis and applications are displayed in perspective.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1988

 оличество страниц: 591

ƒобавлена в каталог: 14.06.2009

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
ѕредметный указатель
Abel convergence test      539Ч540
Abel, reproached by Fourier      532
Adleman, secret codes      505 509
Airy, failures of imagination      30 333
Analytic function (and vice versa)      124 126Ч129 136Ч138
Analytic function Cauchy Riemann equations      121
Analytic function importance in number theory      546Ч556
Analytic function key theorems      379Ч380 382Ч383 397Ч399
Analytic function real part harmonic      122Ч123
Arnold, catastrophe theory      112
Author clad in a little brief authority      584
Author clad in nothing      395
Bachelier, economic Brownian motion      582Ч584
Bell, invention of telephone      25
Bessel function      174 486Ч487
Brown, discoverer of Brownian motion      43
Brownian motion, economic      581Ч584
Brownian motion, mathematical      see УSeparate entryФ
Brownian motion, physical      43Ч45
Brownian path Levy's theorem      458
Brownian path properties      459
Brownian path, defined      458
Brownian path, used      459Ч460 467Ч469
Burgess Davis, proof of Picard's little theorem      467Ч469
Burt      429Ч441
Cambridge difficulty of exams, in old days      270Ч271
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      111Ч112
Cauchy distribution      246Ч252 583
Cauchy inversion formula, independent discovery of      295
Cauchy Liouville's theorem      175
Central limit theorem proof      347Ч356
Central limit theorem warnings      347 361
Central limit theorem, counter example      251Ч252
Central limit theorem, general form      357Ч361
Central limit theorem, multidimensional      413Ч417
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 УTchebychevФ
Chi squared distribution      415 423
Chi squared test      418Ч424 436Ч438
Clarke, A.M. and A.D.B., on Burt      435 440Ч441
Code making general criteria      503Ч505
Code making proposal of Rivest, Shamir and Adleman      509Ч511
Codebreaking, English Civil War      504Ч505
Codebreaking, World War II      92 503
Compass, Kelvin's work on      28Ч29 97
Complex variable      see УAnalytic functionsФ
Computers, early      30Ч31 47 92
Constructions by piling up of functions, divergent Fourier sum      67Ч73
Constructions by piling up of functions, nowhere differentiable function      38Ч41
Constructions by piling up of functions, sliding hump      73
Constructions by piling up of functions, unpleasant convolution      570Ч572
Constructions by piling up of functions, unpleasant Laplace transform      383Ч384
Convolution on Euclidean space natural occurrence      413Ч414 484Ч485
Convolution on Euclidean space, defined      412
Convolution on roots of unity, introduced      495Ч496
Convolution on roots of unity, neat trick, using      501Ч502
Convolution on the circle elementary properties      259Ч261
Convolution on the circle, defined      259
Convolution on the real line      see also УKernelsФ
Convolution on the real line counter example      570Ч572
Convolution on the real line in proof of Weierstrass's theorem      292Ч294
Convolution on the real line key slogan      262
Convolution on the real line properties      253Ч258 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      565Ч572
Cooley, Fast Fourier Transform      499
Coprimality, probability of      528Ч531
Crystallography      481Ч483
Damped oscillator Duffing's equation      101Ч115
Damped oscillator stability      79Ч98
Damped oscillator, defined      79
Darwin golden rule      433
Darwin on Brown      43
Darwin Origin of Species      283 289Ч291
Data, how not to present      439
Data, how not to present (table)      430
Data, how to present      433
De Moivre's theorem      see УCentral limit theoremФ
Demon of Chance, presides over financial markets      584
Differentiation and convolution      263 265Ч266 269
Differentiation under the integral sign      265Ч269
Dirichlet convergence of Fourier sums      3
Dirichlet encouraged by Fourier      480
Dirichlet L-function, defined      540
Dirichlet L-function, properties      537Ч538 541Ч550
Dirichlet problem      see also УLaplace's equationФ
Dirichlet problem for disc      121Ч123
Dirichlet problem uniqueness      124Ч125 135
Dirichlet problem with smoothness assumptions      124Ч130
Dirichlet problem without smoothness assumptions      131Ч141
Dirichlet problem, existence problem (partial solution)      125
Dirichlet problem, existence problem (possible non-existence)      139Ч140 140 163Ч164
Dirichlet problem, Monte Carlo method      47Ч49
Dirichlet theorem on primes in arithmetic progression      532Ч557
Dorfman, on Burt      435 438Ч439
Double periodicity, for analytic functions      175
Double tides      24
du Bois-Reymond, divergence of Fourier sums      3 67Ч73
Duffing's equation      101Ч115
Egyptology, Fourier's importance in      477
Eigen functions and values      181
Einstein special relativity      291
Einstein, Brownian motion      45
Ellipsoid, rotations of      416Ч417 420Ч421
Energy arguments for existence of solution of Dirichlet's problem      125
Energy arguments for existence of solution of Dirichlet's problem (and for uniqueness)      124Ч125
Energy arguments leading to Liapounov's method      79Ч81 84
Energy arguments, problems with      126 131Ч133 140 163Ч164
Equidistribution      11Ч14 47 408
Equiripple criterion, for best uniform approximation      202Ч206 212 215
Euclid algorithm      507
Euclid infinite number of primes      271 507 527
Euler Dirichlet formula      534Ч535
Euler generalisation of Fermat's little theorem      508
Euler remarkable formula      525Ч531
Euler remarkable formula (extended by Dirichlet)      532Ч535
Euler totient function      507Ч508
Euler, anticipates Fourier      170
Euler, infinite number of primes      527Ч528
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      473Ч474
Excessive pessimism, general      4 74
Excessive pessimism, Lagrange      473
Excessive pessimism, Tchebychev      198
Eysenck, and Burt      429Ч432 434Ч435
Faraday      271 273 332Ч333
Fast factorisation no general method known      505 511
Fast factorisation work of Lenstra      512
Fast Fourier Transform      497Ч499
Fast multiplication      500Ч503
Fejer      4Ч5
Fejer's theorem for Fourier transforms      223Ч225 240Ч243
Fejer's theorem proof and elementary consequences      6Ч10
Fejer's theorem, discussed      4Ч5
Fermat's little theorem      508
Fermi, Monte Carlo method      46
Fields, transatlantic cable      332Ч336
Fisher      424Ч425 427 439
Fourier analysis on groups, finite groups      513Ч518
Fourier analysis on groups, roots of unity      491Ч496
Fourier analysis on groups, techniques of book inadequate for non-Abelian groups      517Ч518
Fourier analysis on groups, techniques of book inadequate for non-Abelian groups (but work well for finite Abelian groups)      519Ч524
Fourier coefficients and convolution      259Ч261
Fourier coefficients Parseval's formula      156 183
Fourier coefficients uniqueness      9
Fourier coefficients, behaviour      153 260Ч261 409
Fourier coefficients, defined      3
Fourier discovers fundamental heat equation      25 478
Fourier generalises Fourier sums to non-trigonometric case      171Ч174
Fourier letter to Lagrange      170Ч171
Fourier life      475Ч480
Fourier linear programming      480 499
Fourier on behaviour of solutions of differential equations      88
Fourier sums      see also УSumsФ
Fourier sums best mean square approximation      148Ч149
Fourier sums Carleson's convergence theorem      75
Fourier sums Dirichlet's convergence theorem      3 61
Fourier sums divergence      3 67Ч75
Fourier sums near discontinuity      59Ч66 411
Fourier sums quality of approximation      28 35Ч37 151Ч152
Fourier sums rate of convergence      35Ч37 151Ч152
Fourier sums simplest convergence theorem      32Ч34 153 408 411
Fourier sums wider convergence theorem      57Ч58
Fourier sums, differentiation of      33 38
Fourier sums, multidimensional      409Ч411
Fourier sums, non-trigonometric (Sturm Liouville)      173Ч174 179Ч187
Fourier temperature of subsoil      25Ч27
Fourier transforms and convolution      257Ч258
Fourier transforms behaviour      263
Fourier transforms elementary properties      221Ч223
Fourier transforms heuristics      221
Fourier transforms in probability      245Ч252 347Ч356
Fourier transforms inversion theorem      295Ч298
Fourier transforms inversion theorem (with discontinuities)      300Ч307
Fourier transforms uniqueness      244
Fourier transforms, defined      221 580
Fourier transforms, multidimension      411Ч412
Fourier transforms, natural occurrence in optics      481Ч483
Fourier transforms, numerical calculation needed      481Ч487
Fourier transforms, numerical calculation of      488Ч490 497Ч499
Fourier views on mathematics      532 558Ч559
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      191Ч196
Gibbs phenomenon      62Ч66
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 УFourier analysis on groupsФ
Groups of units modulo      506Ч508
Groups structure theorem for finite Abelian      519Ч524
Hadamard, shows failure of energy arguments      131Ч133
Haldane, life and opinions      425Ч428
Hardy keeps Titchmarsh in the dark      481
Hardy on English isolation      370
Hardy rejoices in remoteness of number theory      509Ч510
Hardy, Lebesgue measure essential for Fourier theory      572
Harmonic analyser      30Ч31 62
Harmonic function (function satisfying Laplace's equation)      see УUnder that entryФ
Hausdorff, moment theorem      21Ч22 385
Heat equation and age of earth      285Ч288 370
Heat equation discovered by Fourier      25 478
Heat equation for cooling sphere      171Ч174
Heat equation for infinite rod      274Ч281
Heat equation for semi-infinite rod      287Ч288 308Ч323
Heat equation for temperature of subsoil      25Ч27
Heat equation for transatlantic cable      333Ч336
Heat equation solution by convolution      277Ч281
Heat equation solution by Fourier transforms      274Ч277
Heat equation uniqueness of solutions      277 344Ч346
Heat equation uniqueness of solutions (and non-uniqueness)      338Ч343
Heaviside, opinions      370Ч371 406
Helmholtz and Kelvin      336
Helmholtz on linearity      24Ч25
Herivel, biography of Fourier      170 475
Humboldt, patron of Dirichlet and Jacobi      556
Huxley, УGarbage inФ, elegant formulation of      290
Images, method of      286Ч287
Inequalities for Fourier coefficients      153
Inequalities, Bessel      150
Inequalities, Cauchy      152
Inequalities, Cauchy, Schwartz, Buniakowski      146 152 180 492
Inequalities, isoperimetric      see УIsoperimetric problemФ
Inequalities, Tchebychev      47 251Ч252
Inequalities, triangle      147 181 492
Inner product and geometric intuition      147Ч148
Inner product associated with Sturm Liouville      180
Inner product on roots of unity      491Ч496
Inner product with weights      185
Inner product, discussed      145Ч158
Instability      see УStabilityФ
Integration, numerical      46Ч47 191Ч196
Interchange of integrals easy for finite ranges      226Ч227
Interchange of integrals for infinite ranges      229 233Ч239
Interchange of integrals, counter examples      226 231
Interchange of sums, counter example      230Ч231
Interchange of sums, justified      231Ч233
Inversion theorems      see УFourier transformФ УLaplace
Isoperimetric problem solution of Hurwitz via Fourier series      166Ч169
Isoperimetric problem solution of Steiner via geometry      159Ч163
Isoperimetric problem, discussed      159 164Ч165
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      283Ч291
Kelvin, Lord (Thomson), and Helmholtz      336
Kelvin, Lord (Thomson), founder of thermodynamics      271Ч272
Kelvin, Lord (Thomson), grandfather of computer      30Ч31
Kelvin, Lord (Thomson), intellectual relation to Faraday, Fourier, Green, Liouville and Maxwell      270Ч273
Kelvin, Lord (Thomson), invents mirror galvanometer      334
Kelvin, Lord (Thomson), leg pulled by Maxwell      575
Kelvin, Lord (Thomson), life      270Ч273 336Ч337
Kelvin, Lord (Thomson), method of images      286
Kelvin, Lord (Thomson), redesigns compass      28Ч29 97
Kelvin, Lord (Thomson), Thomson and Tait      273
Kelvin, Lord (Thomson), tidal prediction      29Ч31
Kelvin, Lord (Thomson), transatlantic cable      333Ч337
Kendall, economic Brownian motion      584
Kernels and convolution      254Ч255 261
Kernels as approximate identities      261
Kernels heat      279 292Ч293
Kernels on roots of unity      493Ч496
Kernels, Dirichlet      68Ч69
Kernels, Fejer      6Ч8 195 407
Kernels, modified Fejer      240Ч242
Kernels, Poisson      118Ч119
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      537Ч538 541Ч550
L-function, defined      540
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