Note on Fourier’s Theorem

Extract

Professor Bryan’s paradox on Fourier’s theorem (Gazette, vol. iv. p. 390) seems to call for a few words of explanation.

In the first place a function f(θ) is constructed of the form

and the function is arranged so as to have the same values as an arbitrary function F(θ) at the places θ = 0, a, 2a, …, (n-1)a, where α=2Π/n. It will be observed here that we have only n data from which to determine (2n-1) coefficients, and consequently the form of f(θ) is largely at our disposal ; but the particular form selected by Professor Bryan leads to the formulae for the coefficients,