Cosine Products, Fourier Transforms, and Random Sums

one gets the first actual formula for 7r that mankind ever discovered, dating from 1593 and due to Frangois Viete (1540-1603), whose Latinized name is Vieta. (Was any notice taken of the formula's 400th anniversary, perhaps by the issue of a postage stamp?) From the samples of a function t(x) at equally spaced points x", n E z, one can reconstruct the complete function with the aid of sin x/x, provided t is "band-limited" and the spacing of the samples is small enough. This is the content of the Sampling Theorem, which lends its name to sin x/x as the sampling function. Its importance in signal processing, where it is also known as sinc x, is the result of its Fourier transform being the characteristic function of the interval [-1, 1] (modulo a scalar factor). In Section 2 we prove the infinite product expansion for sin x/x and derive Viete's formula. In Section 3 we transform the product expansion with the Fourier transform and use convolution and delta distributions to prove it in a way that reveals a host of similar identities. Section 4 puts these identities into a probabilistic setting, and in Section 5 we alter the probability experiments in order to make connections between infinite cosine products, Cantor sets, and sums of series with random signs, particularly the harmonic series. This leaves us with some interesting unsolved problems and conjectures for further work.