A survey of some of the main-line developments in trigonometric series. It is an extension of a Presidential address to Section III of the Royal Society of Canada. Part I deals briefly, in chronological order, with the mathematical problems that arose out of trigonometric series and the interactions between these problems and general mathematical analysis. There is mention of the contributions of Euler, Daniel Bernoulli, d'Alembert, Fourier, Lagrange, Dirichlet. This is followed by a description of some of the problems still open at the end of the period covered by these men, and a description of the way these problems were later solved by Denjoy, by Zygmund and Marcinkiewicz, by Burkill and by James. Part II gives complete proofs of the results of the earlier period outlined in Part I, and concludes with the essential details of the approach of Denjoy and of James.

Canadian Mathematical Congress Lecture Series, No. 2.