Carleson's theorem

Statement

For every square-integrable periodic function $f \in L^2([-\pi, \pi])$, the symmetric partial sums $S_N f(x) = \sum_{n=-N}^{N} \hat{f}(n) e^{inx}$ of its Fourier series, where $\hat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int}\,dt$, converge pointwise to $f(x)$ as $N \to \infty$ for Lebesgue-almost every $x \in [-\pi, \pi]$. (Extended by Richard Hunt in 1968 to every $f \in L^p([-\pi, \pi])$ with $1 < p < \infty$.)

Why is it true?

Parseval's identity guarantees that the Fourier partial sums $S_N f$ converge to $f$ in average $L^2$ energy, so Nikolai Luzin conjectured in 1915 that $S_N f(x)$ should also converge at almost every individual point $x$. Yet Andrey Kolmogorov shocked analysts in 1923–1926 by constructing an integrable function $f \in L^1([-\pi, \pi])$ whose Fourier series diverges everywhere, making many suspect Luzin's conjecture was false even for $L^2$. Lennart Carleson proved in 1966 that finite energy $f \in L^2([-\pi, \pi])$ is strong enough to tame the wild oscillations of the Dirichlet kernel: destructive interference across scales prevents the partial sums from blowing up except on a set of measure zero.

Topics that use this theorem

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Step-by-step proofs

No step-by-step proof yet for this theorem.