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Korner T.W. Ч Exercises in Fourier Analysis
Korner T.W. Ч Exercises in Fourier Analysis

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Ќазвание: Exercises in Fourier Analysis

јвтор: Korner T.W.


A collection of exercises on Fourier analysis that will thoroughly test the understanding of the reader is arranged chapter by chapter to correspond with An Introduction to Fourier Analysis. For all who enjoyed that book, this companion volume will be an essential purchase.

язык: en

–убрика: ћатематика/

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

ed2k: ed2k stats

√од издани€: 1993

 оличество страниц: 395

ƒобавлена в каталог: 05.04.2008

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ѕредметный указатель
Abel, behaviour on circle of convergence      67 100
Abel, brilliant stroke      155
Abel, convergence test      67
Abel, integral equation      277 281Ч283
Abel, Summation Formula      66 80
Abel, theorem on product of series      100
African Eve      371
Almost periodic functions      36Ч38 133
Apery, 'Miraculous demonstration'      172Ч173
Approximation      see "Rate of convergence"
Bachelier, vindicated      375Ч377
Band limited functions      231Ч235 241Ч243
Bernoulli justifiable pride      69
Bernoulli numbers      68Ч71 78 222Ч223 263Ч264 see
Bernoulli polynomials      68Ч71 73Ч74 76
Bernstein approximation theorems      41Ч43 54
Bernstein inequalities      10 176 188Ч189 240Ч241
Bernstein polynomials      25Ч26 246
Bessel functions      321Ч325
beta function      273Ч275
Beta function, incomplete, Phillip Hall's view of      275
Beukers, proof of Apery's result      173Ч175
Black boxes      226Ч229 294Ч297
Blown up, Gibbs phenomenon      59
Bounded variation, functions of      55Ч58 204Ч205
Brent and Salamin, computation of pi      333Ч334
Brownian motion, construction of      366Ч369
Brownian motion, Fourier series of      372
Brownian motion, Levy's theorem on uniqueness of      374Ч377
Brownian motion, nowhere differentiable      371Ч372
Byrnes, hosts hostile tribes      242
Cantor, existence of transcendentals      144Ч145
Cantor, uniqueness theorem      111Ч113
Caratheodory's lemma for convex sets      185
CAT (computer assisted tomography)      318Ч319
Causality conditions      229 234Ч236 294
Central limit theorem      51
Champagne, good, offer of      139
Change of variable for Fourier transform      314
Change, how to count      259Ч262
Chebychev      see "Tchebychev"
Compact discs (articles of commerce)      234
Concave function, Jensen's inequality for      124Ч125
Convergence of Fourier sums      55Ч58 200 203Ч205
Convex function      see "Concave function"
Convex sets      122Ч123 185Ч186
Crinkles in nature      140 143
Cyber-punk Tripos question      371
de la Vallee Ч Poussin, prime number theorem      358Ч365
De Moivre's theorem      78Ч80
Depression, text book induced      151
Dido's problem      218
Difference equation, linear, how to solve      13Ч14
Differential equations, elementary functions, used to define      88Ч91
Differential equations, existence of solutions for      85Ч89
Differential equations, linear, how to solve      15 248Ч249
Differential equations, numerical solution of      93Ч94
Differential equations, rate of growth of solutions for      91Ч93
Dirichlet convergence test      66
Dirichlet evaluation of Gauss sum      329
Dirichlet lemma on approximation      34Ч35 299
Dirichlet original proof of convergence of Fourier sums      203Ч205
Dirichlet pigeonhole principle      35
Dirichlet problem, Neuman problem, related to      107Ч110
Dirichlet problem, obviously not soluble on physical grounds      116Ч117
Dirichlet problem, obviously soluble on physical grounds      115Ч116
Dirichlet problem, punctured plane, solution for      106
Dirichlet series      352 356Ч357
Divergent Fourier series, Fejer's example      80Ч81
Divergent Fourier series, Kahane and Katznelson, theorem of      82Ч84
Divergent Fourier series, Kolmogorov's everywhere divergent example      301Ч305
Divergent Fourier series, rate of divergence      64Ч66 202
Divergent Fourier transform      190
Doubly periodic functions      see "Elliptic functions"
Eigenvalues via minimisation      157Ч163
Elliptic functions, properties and constructions      147Ч155 236Ч239
Elliptic functions, why so called      155
Elliptic integrals, equidistribution      11Ч13 15Ч16 130Ч133
Elliptic integrals, link with elliptic functions      155
Elliptic integrals, spectacular use of      332Ч334
Euler conjectures law of quadratic reciprocity      339
Euler equality (four squares)      300Ч301
Euler formula for Bernoulli numbers      70 222Ч223 263Ч264
Euler gamma      292 349 350 354
Euler infinite product for sine      285Ч286
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 222Ч223
Euler Ч Maclaurin summation formula      61Ч62 71Ч77 131 350 353Ч354
Factorial function      see "Gamma function"
Feedback      see "Black boxes"
Fejer      11 80Ч81 181Ч183
Feller, proof of Stirling's formula      290Ч291
Feynman, anecdote      207
Fractional integral      275Ч277 281Ч283
Function      see "Under specific name"
Gamma function, and Beta function      273Ч274
Gamma function, definition      270
Gamma function, duplication formula      271Ч273 288Ч290
Gamma function, factorial, natural extension of      271Ч273
Gamma function, formula involving      277 see "Watson's
Gamma, Euler's      292 349 350 354
Gauss computation of pi      284 331Ч334
Gauss formula for factorial function      272Ч273
Gauss proves law of quadratic reciprocity at age nineteen      338Ч339
Gauss sum      326Ч330 339
Generalised limit      7Ч8
Gibb's phenomenon      59Ч60 63Ч66
Green's function as transfer (or response) function      226Ч229 294Ч299
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      241Ч243
Guiness, does good by stealth      309
Hadamard gap theorem for natural boundaries      266Ч267
Hadamard prime number theorem      358Ч365
Hahn Ч Banach theorem in finite dimensions      123Ч124
Halmos, insufficiently asked question of      39
Hardy and Littlewood inequality      129
Hardy and Littlewood Tauberian theorem      102
Hardy nowhere differentiable function      46Ч47
Hardy test integral      60 208Ч209
Hardy transcendence of gamma, reward offered      349
Harpsichord      97Ч98
Harvard, mathematicians, mannerisms of      45
Hausdorff moment theorem      30Ч31
Hausdorff Ч Young inequality      136Ч139
Heisenberg inequality (uncertainty principle)      230Ч231
Herglotz theorem on positive definite sequences (simple version)      319Ч320
Hermite, quadrature formula      188
High definition TV      50
How not to index a book      381Ч385
How not to interpolate with polynomials      170 179Ч183
How not to throw dice      310
How not to write out a theorem      301 see
How to be almost sure a number is prime      342Ч343
How to build a harpsichord      97Ч98
How to conduct life      318
How to count change      259Ч262
How to expand in partial fractions      255Ч256
How to gamble, if you must      311Ч312
How to solve a linear difference equation      13Ч14
How to solve a linear differential equation      15 184Ч185
How to value stock options      377
Huygens, practical and theoretical work on pendulum clock      281Ч283
Inclusion-exclusion formula      348
Inequality, Bernstein (band limited functions)      240Ч241
Inequality, Bernstein (trigonometric polynomials)      10 176 188Ч189
Inequality, Bessel      133
Inequality, Cauchy (arithmetic-geometric)      124
Inequality, Cauchy, Schwarz, Bumakowski      122
Inequality, Hardy and Littlewood      129
Inequality, Hausdorff Ч Young      136Ч139
Inequality, Heisenberg      230Ч231
Inequality, Hilbert      128Ч129
Inequality, Hoelder      125
Inequality, Hoelder, converse      126
Inequality, Jensen (concave functions)      124Ч125
Inequality, Jensen (zeros of analytic functions)      114Ч115 288
Inequality, Tchebychev (polynomials)      188Ч189
Inequality, Tchebychev (probability)      25 79 197Ч198
Inequality, triangle      122 125 240
Inequality, Van der Corput (first and second)      129Ч131
Inequality, Van der Corput (related to equidistribution)      130Ч131
Inequality, Wirtinger      140 162 239Ч240
Infinite products      77 284Ч291
Insects, love life of      214 302
Interpolation by polynomials      25Ч27 170 178Ч183
Interpolation formula of Rivlin and Shapiro      185Ч189
Jackson approximation theorems      41 43Ч45 54
Jacobi, compulation of Legendre symbol      340Ч342
Jacobi, identity      238Ч239 354Ч356
Jacobi, remainder in Euler Ч Maclaurin formula      73
Jacobi, symbol      340Ч342
Jacobi, theta function      236Ч239
Jigsaw puzzle, mathematician's      231
Kahane and Katznelson, divergent Fourier series      82Ч84
Kahane, beautiful probabilistic arguments      261 371Ч372
Karamata, proof of Tauberian theorem      102Ч103
Katznelson      see "Kahane and Katznelson"
Kolmogorov everywhere divergent Fourier series      301Ч305
Kolmogorov zero-one law      267Ч9
Kramers Ч Kronig (causality) relations      234Ч236
Kronecker's theorem (on simultaneous approximation), another proof      337
Kronecker's theorem (on simultaneous approximation), used      301Ч305
Lacunary series      266Ч270
Lagrange's four squares theorem      300Ч301
Landau driven out of Gottingen      91
Landau sage advice      41
Laplace transforms, convolution, for      257Ч258
Laplace transforms, delicate treatment of      363Ч364
Laplace transforms, partial differential equations, used with      258
Laplace transforms, warning      93 see
Lattes, remarkable rational function      153Ч154
Lebesgue, proof of Weierstrass theorem      27Ч28
Lebesgue, thorn      115Ч117
Legendre symbol      338Ч343 345Ч346
Lerch's theorem, by complex variable      287 288
Levy's theorem on uniqueness of Brownian motion      374Ч376
Lipschitz condition      87Ч88
Littlewood      see also "Hardy and Littlewood"
Littlewood, obiter dicta      138 301
Matrices, fun with      95Ч96 157Ч158 248Ч251 301 307Ч308 314 327Ч328
Mean value theorems for integrals      67 202Ч203
Mellin transform      343Ч344 351Ч352
Minkowski, inequality      125
Minkowski, theorem on geometry of numbers      299Ч301
Monte Carlo method      51Ч52
Mordell, neat evaluation      198
Mordell, proof of Hadamard gap theorem      266Ч267
Mordell, suitable thesis topic      349
Natural boundary for analytical functions      264Ч270 286
Neumann problem      107Ч110
Newman D.J., makes hard things less hard      333Ч334 363Ч364
Newton Ч Cotes formula, problems with      169Ч70 278Ч279
Newton, sums of powers of roots      15Ч16
Nowhere differentiate functions, Brownian motion      371Ч372
Nowhere differentiate functions, Hardy      46Ч47
Nowhere differentiate functions, Van der Waerden      48
Nowhere differentiate functions, Weierstrass      46Ч47
Numerical analysts, wise sayings of      43
Numerical quadrature (integration)      168Ч170 180 278Ч279
Nyquist, criterion for feedback      296Ч297
Nyquist, rate for signal transmission      231Ч232 241Ч242
Oven, microwave, unsatisfactory      33
Pal, interpolation by polynomials with integer coefficients      26Ч27
Paley Ч Wiener type theorem      234
Parseval's equality for Fourier transforms      224 240Ч241
Partial fractions, how to get      255Ч256
Partial fractions, not as boring as they look      260Ч261
Peano, kernel theorem      21Ч23
Peano, space filling curve      49Ч50
Pendulum clock      281Ч283
Periodicities, hidden      34
Pi, computation of      77 284 331Ч334
Pisot number      15Ч16
Poisson, formula      61 222 231 239
Poisson, summation      100 103Ч105
Polya, infinity of primes      211
Polya, numerical quadrature      169Ч170 278Ч279
Polya, random walks      312Ч313
Polynomial, Bernoulli      68Ч71 73Ч74 76
Polynomial, Bernstein      25Ч26 246
Polynomial, Laguerre      166Ч167
Polynomial, Legendre      164Ч166 175
Polynomial, Rudin Ч Shapiro      138Ч139
Polynomial, Tchebychev      25 105 168Ч169 188Ч189
Polynomials, interpolation by      25Ч27 170 178Ч183
Polynomials, sums of powers of roots      15Ч16
Population, explosion      370
Population, oscillation      251Ч253
Positive definite quadratic forms      95Ч96
Positive definite quadratic sequencies      319Ч320
Potatoes, two, and string, what to do with      98Ч99
Prime number theorem, full, of Hadamard and de la Vallee Ч Poussin, proved      358Ч365
Prime number theorem, full, of Hadamard and de la Vallee Ч Poussin, used      173Ч175
Prime number theorem, limited, of Tchebychev, proved      171Ч173
Prime number theorem, limited, of Tchebychev, used      343 362Ч363
Prime numbers, probabilistic test for      342Ч343
Primes, infinity of Euclid      211
Primes, infinity of Polya      211
Radar, electronically steered      315Ч317
Radon transform      318Ч319
Random series, construction of Brownian motion using      366Ч369
Random series, example given by Fourier series of Brownian motion      372Ч374
Random series, Taylor, and natural boundaries      269Ч270
Random walk returns      312Ч313
Rate of convergence of approximations      41Ч45 53Ч54
Reconstruction of band limited signals      231Ч234 241Ч243
Response function (transfer function)      226Ч229 294Ч297
Richardson L.F., deferred approach to the limit      73 75
Riemann integral, unsatisfactory      127Ч128 135
Riemann Ч Lebesgue lemma      10 200Ч201 220
Riemann, Fourier series, work on      111Ч113
Riemann, fractional integrals      275Ч279
Riemann, functional equation for zeta      354Ч356
Riemann, hypothesis      361 364Ч365
Riemann, localisation (weak version)      201
Riemann, Mapping Theorem      118Ч121
Riemann, thesis report on      121
Riemann, zeta function      see "Main entry"
Rivlin and Shapiro interpolation formula      185Ч189
Salamin      see "Brent and Salamin"
Schur, I., evaluation of Gauss sum      326
Schur, I., proof of two inequalities of Hilbert      128Ч129
Schur, I., theorem on making change      259Ч261
Set theory, origin of      113
Shannon sampling theorem      231Ч234
Shapiro      see also "Rivlin and Shapiro"
Shapiro, MSc thesis      138
Simpsin's rule, error bounds for      22Ч23
Smith, genealogy applied to Brownian motion      371Ч372
Smith, why are we all called      371
Smith, why are we not all called      369Ч371
Smoke rings      216
Solvay and Strassen's probabilistic test for primes      342Ч343
Space filling curve of Liu Wen      49Ч50
Sphere, heating and cooling      32Ч33 216Ч218
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