Carleson’s theorem asserts that the FOURIER SERIES [III.27] of a function f in L2[0, 2π] converges almost everywhere. To understand this statement and appreciate its significance, let us follow the history of the subject, starting in the early nineteenth century. FOURIER’S [VI.25] great idea was that “any” (complex-valued) function f on an interval such as [0, 2π] can be expanded in what we would now call a Fourier series,
for suitable Fourier coefficients an. Fourier obtained the formula for the coefficients an, and proved that (1) holds in interesting special cases.
The next major advance, due to