Fourier Techniques and Applications

The first systematic methods of Fourier analysis date from the eighteenth century with the work of Joseph Fourier on the problems of the flow of heat. Briefly, Fourier noticed that the solution of the heat diffusion problem was simple if the initial temperature distribution was sinusoidal. He then asserted that any distribution can be decomposed into a sum of sinusoids, these being the harmonics of the original functions. This meant that the general solution could now be obtained by summing the solutions of the component sinusoidal problems.This remarkable observation has led to the routine use of the methods in a great diversity of areas - astrophysics, computing, economics, electrical engineering, geophysics, information theory, medical engineering, optics, petroleum and mineral exploration, quantum physics and spectroscopy, to name a few.With this in mind it was decided to have a week of applied Fourier Analysis at the University of New South Wales with the idea of bringing together a wide cross-section of users of Fourier and spectral methods and mathematicians interested in helping to develop and refine new methods.The week was in two parts. Part 1 was a two-day course entitled "Applied Fourier Analysis" and Part 2 was a conference entitled "Fourier Techniques and Applications" in which 25 papers were presented. This volume contains the lecture notes for the two-day course and Part II five of the invited papers and of another two.ContentsThe life of Joseph Fourier (R. N. Bracewell)PART 1: APPLIED FOURIER ANALYSISLinear Systems, Filters and Convolution Theorems (J.W. Sanders)Uncertainty Principles and Sampling Theorems (J.F. Price)Fourier Techniques in Two Dimensions (R.N. Bracewell)Some Mathematical Methods for Spectrum Estimation (J.J. Benedetto)Some Practical and Statistical Aspects of Filtering and Spectrum Estimation (T.P. Speed)PART 2: FOURIER TECHNIQUES AND APPLICATIONSTwo-Dimensional Phase Restoration (R.H.T. Bates and W.R. Fright)Fourier Uniqueness Criteria and Spectrum Estimation Theorems (J.J. Benedetto)Harmonic Synthesis - Theoretical Bounds (G. Brown)Seismic Image Processing for Petroleum Exploration (Abstract) (P.W. Buchen)Fourier Theory in Modern Imaging (T.W. Cole)An Overview of Time and Frequency Limiting (H.J. Landau)