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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.5 Carleson’s Theorem

Charles Fefferman

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. FOURIERS [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,

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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

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