The evolution of applied harmonic analysis. Models of the real world. With a foreword by Ronald N. Bracewell (Q1770213)

For a little more than fifty years abstract modern harmonic analysis has evolved in ways independent of physical applications. It has become intertwined with abstract group theory and a host of modern analytical techniques. The broader subject has a much longer history, however, as a branch of applied mathematics which dates back to the active period of the eighteenth century. The present book is concerned with this longer history as an applied science. In each of the application areas the author provides a sketch of the underlying mathematical tools borrowed from harmonic analysis. The first area is that of historical origins which dealt with waves and heat diffusion. In 1755 Daniel Bernoulli, pursuing work begun earlier by his father, boldly stated that every motion of a string is expressible in the familiar form as a trigonometric series. Some years later, Jean Fourier considered the problem of heat diffusion in continuous bodies such as planar lamina with boundaries held at constant temperature. He derived the familiar partial differential heat equation and found solutions in the familiar form \(\exp(nx) \cos(ny)\) and claimed that the general solution was given by a combination of these with arbitrary coefficients. Fourier also returned to the problem of the vibrating string and was able to ``confirm Daniel Bernoulli's opinion''. This fundamental discovery, that various functions can be represented by trigonometric series, has sparked so many applications over the years. The author discusses the topics of the fast Fourier transform, signal sampling, quantization, and filtering that are mainstays of modern digital communications as applied to telecommunications and astrophysics. The topics of sound and light provide graphic demonstrations of the role of the spectrum and higher dimensional filtering. Perhaps the most important of all applications are the recent ones to medical technology. One example is provided by computerized axial tomography (CAT scans), which are two dimensional images (density distributions) resulting from measuring the attenuation of x-rays along different paths and rotational angles perpendicular to the vertical axis of the human body. Building these images as well as combining them to obtain three dimensional reconstructions requires harmonic analytical transform techniques.