The first systematic methods of Fourier analysis date from the early eighteenth century with the work of Joseph Fourier on the problem of the flow of heat. (A brief history is contained in the first paper.) Given the initial temperaยญ ture at all points of a region, the problem was to determine the changes in the temperature distribution over time. Understanding and predicting these changes was important in such areas as the handling of metals and the determination of geological and atmospheric temperatures. Briefly, Fourier noticed that the solution of the heat diffusion problem was simple if the initial temperature disยญ tribution was sinusoidal. He then asserted that any distriยญ bution can be decomposed into a sum of sinusoids, these being the harmonics of the original function. This meant that the general solution could now be obtained by summing the soluยญ tions of the component sinusoidal problems. This remarkable ability of the series of sinusoids to describe all "reasonable" functions, the sine qua non of Fourier analysis and synthesis, has led to the routine use of the methods originating with Fourier in a great diversity of areas - astrophysics, computing, economics, electrical engineering, geophysics, information theory, medical engineering, optics, petroleum and mineral exploration, quanยญ tum physics and spectroscopy, to name a few
CONTENT
The Life of Joseph Fourier -- 1: Applied Fourier Analysis -- Linear Systems, Filters and Convolution Theorems -- Uncertainty Principles and Sampling Theorems -- Fourier Techniques in Two Dimensions -- Some Mathematical Methods for Spectrum Estimation -- Some Practical and Statistical Aspects of Filtering and Spectrum Estimation -- 2: Fourier Techniques and Applications -- Two-Dimensional Phase Restoration -- Fourier Uniqueness Criteria and Spectrum Estimation Theorems -- Harmonic Synthesis โ{128}{148} Theoretical Bounds -- Seismic Image Processing for Petroleum Exploration (Abstract) -- Fourier Theory in Modern Imaging -- An Overview of Time and Frequency Limiting -- From Rainbows to Rings: A History of the Idea of the Spectrum (Outline) -- List of Contributors
