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Korner T.W. — Fourier Analysis
Korner T.W. — Fourier Analysis



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Íàçâàíèå: Fourier Analysis

Àâòîð: Korner T.W.

Àííîòàöèÿ:

Every mathematician should have this book. It covers the ideas, techniques and results to be obtained from Fourier Analysis. Also covers applications in number theory, statistics, earth science, astronomy and electrical engineering.
Körner provides a shop-window for some of the ideas, techniques and elegant results of Fourier analysis.


ßçûê: en

Ðóáðèêà: Ìàòåìàòèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1988

Êîëè÷åñòâî ñòðàíèö: 608

Äîáàâëåíà â êàòàëîã: 18.08.2009

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
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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, Laplace transform is analytic      379—382
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, defined      458
Brownian path, Levy's theorem      458
Brownian path, properties      459
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 used to construct Brownian motion      50
Central limit theorem, counter example      251—252
Central limit theorem, general form      357—361
Central limit theorem, multidimensional      413—417
Central limit theorem, proof      347—356
Central limit theorem, warnings      347 361
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, defined      412
Convolution on Euclidean space, natural occurrence      413—414 484—485
Convolution on roots of unity, introduced      495—496
Convolution on roots of unity, neat trick, using      501—502
Convolution on the circle, defined      259
Convolution on the circle, elementary properties      259—261
Convolution on the real line in proof of Weierstrass's theorem      292—294
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, counter example      570—572
Convolution on the real line, defined      253
Convolution on the real line, key slogan      262
Convolution on the real line, more general      565—572
Convolution on the real line, properties      253—258 265 see
Cooley, Fast Fourier Transform      499
Coprimality, probability of      528—531
Crystallography      481—483
Damped oscillator, defined      79
Damped oscillator, Duffing's equation      101—115
Damped oscillator, stability      79—98
Darwin on Brown      43
Darwin Origin of Species      283 289—291
Darwin, golden rule      433
Data, how not to present      430 439
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 problem for disc      121—123
Dirichlet problem with smoothness assumptions      124—130
Dirichlet problem without smoothness assumptions      131—141 see
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 problem, uniqueness      124—125 135
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, 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, Brownian motion      45
Einstein, special relativity      291
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, anticipates Fourier      170
Euler, generalisation of Fermat's little theorem      508
Euler, infinite number of primes      527—528
Euler, remarkable formula      525—531
Euler, remarkable formula, (extended by Dirichlet)      532—535
Euler, totient function      507—508
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, discussed      4—5
Fejer's theorem, proof and elementary consequences      6—10
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 behaviour      153 260—261 409
Fourier coefficients Parseval's formula      156 183
Fourier coefficients uniqueness      9
Fourier coefficients, defined      3
Fourier discovers fundamental heat equation      25 478
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, best mean square approximation      148—149
Fourier sums, Carleson's convergence theorem      75
Fourier sums, differentiation of      33 38
Fourier sums, Dirichlet's convergence theorem      3 61
Fourier sums, divergence      3 67—75
Fourier sums, multidimensional      409—411
Fourier sums, near discontinuity      59—66 411
Fourier sums, non-trigonometric (Sturm Liouville)      173—174 179—187
Fourier sums, quality of approximation      28 35—37 151—152
Fourier sums, rate of convergence      35—37 151—152 see
Fourier sums, simplest convergence theorem      32—34 153 408 411
Fourier sums, wider convergence theorem      57—58
Fourier temperature of subsoil      25—27
Fourier transforms and convolution      257—258
Fourier transforms in probability      245—252 347—356
Fourier transforms, behaviour      263
Fourier transforms, defined      221 580
Fourier transforms, elementary properties      221—223
Fourier transforms, heuristics      221
Fourier transforms, inversion theorem      295—298
Fourier transforms, inversion theorem (with discontinuities)      300—307
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 transforms, uniqueness      244
Fourier views on mathematics      532 558—559
Fourier, generalises Fourier sums to non-trigonometric case      171—174
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 of units modulo n      506—508 see
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 Lebesgue measure essential for Fourier theory      572
Hardy on English isolation      370
Hardy rejoices in remoteness of number theory      509—510
Harmonic analyser      30—31 62
Hausdorff, moment theorem      21—22 385
Heat equation and age of earth      285—288 370
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, discovered by Fourier      25 478
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 on roots of unity      491—496
Inner product with weights      185
Inner product, associated with Sturm Liouville      180
Inner product, discussed      145—158
Instability      see "Stability"
Integration, numerical      46—47 191—196
Interchange of integrals for infinite ranges      229 233—239
Interchange of integrals, counter examples      226 231
Interchange of integrals, easy for finite ranges      226—227
Interchange of sums, counter example      230—231
Interchange of sums, justified      231—233
Inversion theorems      see "Fourier transform" "Laplace
Isoperimetric problem, discussed      159 164—165
Isoperimetric problem, solution of Hurwitz via Fourier series      166—169
Isoperimetric problem, solution of Steiner via geometry      159—163
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) and Helmholtz      336
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), 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 on roots of unity      493—496
Kernels, Dirichlet      68—69
Kernels, Fejer      6—8 195 407
Kernels, heat      279 292—293
Kernels, modified Fejer      240—242
Kernels, Poisson      118—119
Kolmogorov, axioms for probability      428
Kolmogorov, Bachelier and Brownian motion      584
Kolmogorov, divergent Fourier sums      74
Kronecker's theorem, and equidistribution      408
L-function, defined      540
L-function, properties      537—538 541—550
Labeyrie, diameters of stars      484—485
Lagrange, links with Fourier      170—171 476 478—479
Laplace transform and stability      368 388—406
Laplace transform in solution of linear differential equations      377—378
Laplace transform, analyticity      379—382
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