Carleson's theorem
Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of L2 functions, proved by Lennart Carleson (1966). The name is also often used to refer to the extension of the result by Richard Hunt (1968) to Lp functions for p ∈ (1, ∞) (also known as the Carleson–Hunt theorem) and the analogous results for pointwise almost everywhere convergence of Fourier integrals, which can be shown to be equivalent by transference methods.
The result, in the form of its extension by Hunt, can be formally stated as follows:
The analogous result for Fourier integrals can be formally stated as follows:
A fundamental question about Fourier series, asked by Fourier himself at the beginning of the 19th century, is whether the Fourier series of a continuous function converges pointwise to the function.
By strengthening the continuity assumption slightly one can easily show that the Fourier series converges everywhere. For example, if a function has bounded variation then its Fourier series converges everywhere to the local average of the function. In particular, if a function is continuously differentiable then its Fourier series converges to it everywhere. This was proved by Dirichlet, who expressed his belief that he would soon be able to extend his result to cover all continuous functions. Another way to obtain convergence everywhere is to change the summation method. For example, Fejér's theorem shows that if one replaces ordinary summation by Cesàro summation then the Fourier series of any continuous function converges uniformly to the function. Further, it is easy to show that the Fourier series of any L2 function converges to it in L2 norm.
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