## The Evolution of Applied Harmonic Analysis: Models of the Real World"...can be thoroughly recommended to any reader who is curious about the physical world and the intellectual underpinnings that have lead to our expanding understanding of our physical environment and to our halting steps to control it. Everyone who uses instruments that are based on harmonic analysis will benefit from the clear verbal descriptions that are supplied." — R.N. Bracewell, Stanford University A sweeping exploration of essential concepts and applications in modern mathematics and science through the unifying framework of Fourier analysis! This unique, extensively illustrated book describes the evolution of harmonic analysis, integrating theory and applications in a way that requires only some general mathematical sophistication and knowledge of calculus in certain sections. Key features: * Historical sections interwoven with key scientific developments showing how, when, where, and why harmonic analysis evolved * Exposition driven by more than 150 illustrations and numerous examples * Concrete applications of harmonic analysis to signal processing, computerized music, Fourier optics, radio astronomy, crystallography, CT scanning, nuclear magnetic resonance imaging and spectroscopy * Includes a great deal of material not found elsewhere in harmonic analysis books * Accessible to specialists and non-specialists The Evolution of Applied Harmonic Analysis will engage graduate and advanced undergraduate students, researchers, and practitioners in the physical and life sciences, engineering, and mathematics. |

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### Contents

Joseph Fourier The Man and the Mathematician | 1 |

12 The beginnings | 2 |

13 The revolutionary | 4 |

14 At the Ecole Normale | 7 |

15 From imprisonment to the Ecole Polytechnique | 9 |

16 Fourier Napoleon and the Egyptian campaign | 12 |

17 The prefect of Isere and the Analytical Theory of Heat | 16 |

18 Fourier and the Hundred Days | 22 |

58 Storage rings among the biggest machines in the world | 164 |

59 Spectrum of the synchrotron light | 168 |

510 Brightness and undulators | 173 |

511 Applications and the futuristic free electron laser | 174 |

Xray Crystallography Protein Structure and DNA | 179 |

62 Braggs law and the basic role of symmetries | 185 |

63 How to see the invisible | 192 |

64 The phase problem and the direct methods | 196 |

19 Return to Paris | 25 |

Introduction to Harmonic Analysis | 31 |

22 Heat diffusion | 39 |

23 Fourier coefficients and series | 43 |

24 Dirichlet function and theorem | 46 |

25 Lord Kelvin Michelson and Gibbs phenomenon | 49 |

26 The construction of a cellar | 53 |

27 The Fourier transform | 56 |

28 Dilations and the uncertainty principle | 59 |

210 Waves a unifying concept in science | 62 |

Telecommunications and Space Exploration | 65 |

32 More and more hertz | 71 |

33 Tables of Fourier transforms | 76 |

34 The Dirac delta function and related topics | 79 |

35 Convolution | 81 |

36 Filters noise and false alarms | 84 |

37 Sampling and quantization | 87 |

38 The discrete Fourier transform | 91 |

39 The fast Fourier transform | 93 |

history of the FFT | 97 |

311 Unsteady combustion and space propulsion | 101 |

312 The Mars project from von Braun to Korolev and Rubbia | 108 |

Sound Music and Computers | 113 |

ultrasonic reflections in air and water | 116 |

43 Bels decibels and harmonic analysis of a single tone | 117 |

pitch loudness and timbre | 119 |

45 The tempered scale | 122 |

cello clarinet trumpet and bass drum | 124 |

dynamic spectra | 125 |

48 The beginnings of the electronic music synthesizers | 127 |

49 Additive synthesis | 132 |

410 Subtractive synthesis and frequency modulation | 136 |

the vibrating membrane | 138 |

412 Convolution and some special effects | 141 |

413 Computer music and art music | 142 |

Fourier Optics and the Synchrotron Light | 143 |

a close look at new worlds | 147 |

the Fourier transform in two dimensions | 149 |

54 Optical transforms | 152 |

55 Xrays and the unfolding of the electromagnetic spectrum | 157 |

56 Synchrotron radiation first seen in the stars | 160 |

57 Unique features of the synchrotron light | 163 |

65 Brief history of genetics | 197 |

66 DNA through xrays | 202 |

work for a century | 206 |

68 Seeing the hydrogen atom | 210 |

a timeresolved structure | 211 |

The Radon Transform and Computerized Tomography | 215 |

72 Beyond classical radiology | 217 |

73 CT versus MRI | 222 |

74 Projections | 223 |

75 The Radon transform | 224 |

76 Inverting the Radon transform by convolution and back projections | 227 |

77 CT images | 231 |

Nuclear Magnetic Resonance Imaging and Spectroscopy | 237 |

82 Nuclear spin and magnetic moment | 245 |

83 The swing and the phenomenon of resonance | 247 |

the Larmor frequency | 248 |

85 Relaxation process | 250 |

86 Relaxation times | 251 |

chemical shift and coupling constant | 255 |

88 Prion protein | 259 |

continuous wave versus pulses | 261 |

810 The road to NMR imaging in the work of Damadian | 263 |

811 The principle of image formation | 266 |

812 MRIguided interventions | 270 |

813 Functional MRI | 272 |

Radioastronomy and Modern Cosmology | 275 |

92 How radioastronomy was born | 280 |

93 The most mortifying episode of English naval history and radar | 283 |

94 Radio waves and radiotelescopes | 285 |

95 Resolving power | 287 |

96 Interferometry | 290 |

97 The principle of image formation in a radiotelescope | 292 |

98 Aperture synthesis | 294 |

99 Synthesis by Earths rotation | 298 |

910 Quasars pulsars and the Big Bang | 300 |

911 A flat foreverexpanding Universe | 306 |

912 Search for extraterrestrial intelligence | 308 |

Appendix | 311 |

315 | |

337 | |