Math 2.0

Harmonic analysis on Euclidean spaces Convergence of Fourier series Theorem

Carleson's theorem

Builds on Lp spaceFourier seriesFourier coefficientAlmost everywhere

Statement

Let f ⁣:ℝ→ℂf \colon \R \to \C be a 2π2\pi-periodic function that is square integrable on [−π,π][-\pi, \pi], that is, f∈L2(𝕋)f \in L^2(\mathbb{T}) (see Lp space). For k∈ℤk \in \Z let f^(k)=12π∫−ππf(t)e−ikt dt\hat f(k) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-ikt} \, dt be its Fourier coefficients. For N≥0N \ge 0 let SNf(x)=∑k=−NNf^(k)eikxS_N f(x) = \sum_{k=-N}^{N} \hat f(k) e^{ikx} be the partial sums of its Fourier series. Then SNf(x)→f(x)S_N f(x) \to f(x) as N→∞N \to \infty for almost every x∈ℝx \in \R.

Remarks

The proof reduces the theorem to an estimate for the maximal operator Cf(x)=sup⁡N∣SNf(x)∣Cf(x) = \sup_N \abs{S_N f(x)}, namely that it is of weak type (2,2)(2, 2). Convergence holds at every point for trigonometric polynomials, which are dense in L2(𝕋)L^2(\mathbb{T}), and the maximal estimate passes it to all of L2(𝕋)L^2(\mathbb{T}). Carleson proved the maximal estimate in 1966 by a delicate decomposition of the function in time and frequency. Later proofs were given by Fefferman and by Lacey and Thiele.

The theorem settles a conjecture of Lusin. Since continuous periodic functions are square integrable, the Fourier series of a continuous function converges almost everywhere, although it can diverge at a point. The Kahane–Katznelson theorem shows that "almost everywhere" cannot be improved, even for continuous functions. Square integrability cannot be weakened to integrability: Kolmogorov's example is an integrable function whose Fourier series diverges everywhere.

Going deeper

For a student

Convergence in the L2L^2 norm follows from Parseval's identity and is easy. Pointwise convergence is much harder because it needs control of all partial sums at once, through the Carleson maximal operator. Hunt's extension proves the same for LpL^p with 1<p<∞1 < p < \infty. The time-frequency methods behind the proof became a central tool of modern harmonic analysis.

Reading path

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Further reading

  1. Web https://en.wikipedia.org/wiki/Carleson%27s_theorem
About this page
TagFBH6KG484Z
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Kindstatement, theorem
WritingW0 stub
MSC42A20, 42B25
StatusThis copy of the site has no build data. Run m2 status FBH6KG484Z.
AboutFourier series, Convergence of Fourier series
UsesLp space, Fourier series, Fourier coefficient, Almost everywhere
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