Generalized Harmonic Analysis and Tauberian Theorems
Norbert Wiener: Collected Works
The publication of these two exhaustive and definitive papers in book form underlies both their classic nature and their current interest. They retain their considerable power as exercises in pure mathematics. But – and this could not have been anticipated when they were first published in the early '30s in now-inaccessible journals – they have proved to be a prime source of much of today's applied mathematics, especially useful in statistical studies of computation, communication, and control. Wiener himself with characteristic concreteness, here applies his results to the physical problems of continuous spectra and Brownian motions. These applications are not digressive: Wiener knew that if mathematics can aid the sciences, science can enrich mathematics, by suggesting new approaches and lines of attack. As he once wrote, “I grew ever more aware that it was within nature itself that I must seek the language and the problems of my mathematical investigations.”
Wiener described the background and aims of these papers as follows:
Generalized harmonic analysis represents the culmination and combination of a number of very diverse mathematical movements. The theory of almost periodic functions finds its precursors in the theory of Dirichlet series, and in the quasiperiodic functions of Bohl and Esclangon. These latter, in turn, are an answer to the demands of the theory of orbits in celestial mechanics; the former take their origin in the analytic theory of numbers. Quite independent of the regions of thought just enumerated, we have the order of ideas associated with the names of Lord Rayleigh, of Gouy, and above all Sir Arthur Schuster; these writers concerned themselves with the problems of white light, of noise, of coherent and incoherent sources.... Some [expressions of the theory] demand for their proper appreciation a mode of connecting various weighed means of a positive quantity. The appropriate tool for this purpose is the general theory of Tauberian theorems developed by the author.... These latter Tauberian theorems enable us to correlate the mean square of the modulus of a function and the “quadratic variation” of a related function which determines its harmonic analysis.... The Theory of generalized harmonic analysis is itself capable of extensions in very varied directions.... This theory may be developed to cover the case where the infinite sequences of choices is replaced by a haphazard motion of the type known as Brownian.