TABLE OF CONTENTS Is it possible to prove that the definitions, equations, patterns, theorems, that comprise the body of mathematics known as Fourier analysis have value as well as validity? Consider the following:
The function concept; the Riemann and Lebesgue integrals; the concepts of pointwise, uniform, and mean square convergence; Cantor s theory of sets; the Schwartz theory of distributions; were all invented in an attempt to answer the question: When is Fourier s representation valid?, see
E.B.VanVleck, The influence of Fourier s series on the development of mathematics, Science 39(1914), 113 124;
W.A.Coppel, J.B.Fourier On the occasion of his two hundredth birthday, Amer. Math. Monthly 76(1969), 468 483;
S.Bochner, Fourier series came first, Amer. Math. Monthly 86(1979), 197 199;
E.A.Gonz lez-Velasco, Connections in mathematical analysis: The case of Fourier series, Amer. Math. Monthly 99(1992), 427 441.
The most frequently cited mathematics paper ever written,
J.W.Cooley and J.W.Tukey, Math. Comp. 19(1965), 297 301,
describes a clever scheme for doing Fourier analysis on a computer. The FFT, which Gilbert Strang calls the most important algorithm of the 20th century, initiated a revolution in scientific computation. The bibliography in
E.O.Brigham, The Fast Fourier...
