A Century of Interplay Between Taylor Series, Fourier Series and Brownian Motion

Abstract

This article is an extended version of a lecture given in Oxford on 12 May 1995 at the invitation of the London Mathematical Society and the British Society for the History of Mathematics. Contents A few figures Taylor series before 1900. A strange statement of Borel Fourier series before 1900. A strange field Brownian motion around 1900. A rising subject Fourier and Taylor series after 1900. A revival Lacunarity and randomness The appearance of random series of functions The Wiener theory of Brownian motion The merging of Brownian motion and random Fourier series The non-differentiability and local behaviour of Brownian motion Three ways to figure out the Brownian motion The plane Brownian motion Applications of Brownian motion to Taylor series and analytic functions Applications of Brownian motion to Fourier series and harmonic analysis 1991 Mathematics Subject Classification 01A60, 30-03, 30B10, 30B20, 42-03, 42A55, 42A61, 60-03, 60J65.