# Carleson theorem

For a function in $L_2(0,2\pi)$ its trigonometric Fourier series converges almost everywhere. This was stated as a conjecture by N.N. Luzin [1] and proved by L. Carleson [2]. The statement of Carleson's theorem is also valid for functions in $L_p$ for $p>1$ (see [3]). The fact that it does not hold for $p=1$ was shown by an example, constructed by A.N. Kolmogorov [4], of a function in $L_1$ the trigonometric Fourier series of which diverges almost everywhere.

#### References

 [1] N.N. Luzin, "The integral and trigonometric series" , Moscow-Leningrad (1915) (In Russian) (Thesis; also: Collected Works, Vol. 1, Moscow, 1953, pp. 48–212) [2] L. Carleson, "Convergence and growth of partial sums of Fourier series" Acta Math. , 116 (1966) pp. 135–157 [3] R.A. Hunt, "On the convergence of Fourier series" , Proc. Conf. Orthogonal Expansions and their Continuous Analogues , Southern Illinois Univ. Press (1968) pp. 234–255 [4] A. [A.N. Kolmogorov] Kolmogoroff, "Une série de Fourier–Lebesgue divergente presque partout" Fund. Math. , 4 (1923) pp. 324–328

A few years later (than [4]) Kolmogorov anew proved the existence of a function in $L_1$ whose trigonometric Fourier series diverges everywhere [a1].