Harmonic analysis on Euclidean spaces Convergence of Fourier series Theorem

Kahane–Katznelson theorem

Builds on Null set Continuous function Fourier series

Statement

Let $E \subset \R$ be a set of Lebesgue measure zero, that is, a null set. Then there is a continuous $2\pi$ -periodic function $f \colon \R \to \C$ whose Fourier series diverges at every point of $E$ .

Remarks

Kahane and Katznelson proved this in 1966. Together with Carleson's theorem it describes the sets of divergence exactly. A set lies in the divergence set of the Fourier series of some continuous function if and only if it has measure zero. The proof builds $f$ as a uniformly convergent sum of trigonometric polynomials, using covers of $E$ by intervals of small total length. For a single point $E = \set{0}$ the result is the classical example of du Bois-Reymond. For example, $E$ can be the set of rational multiples of $\pi$ , or the Cantor set. Then some continuous function has a Fourier series diverging at each point of $E$ .

Going deeper

For a student

The result shows that Carleson's theorem is sharp for continuous functions. It contrasts with smooth or Hölder continuous functions, whose Fourier series converge everywhere by the Dini test. So the divergence sets of continuous functions are as large as Carleson's theorem allows.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`42DDDZHP7F`
URI	`m2://42DDDZHP7F`
Kind	statement, theorem
Writing	W0 stub
MSC	42A20
Status	This copy of the site has no build data. Run `m2 status 42DDDZHP7F`.
About	Fourier series, Convergence of Fourier series
Uses	Null set, Continuous function, Fourier series