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Charles Louis Fefferman, "Pointwise convergence of Fourier series", Ann. of Math. 98 (1973), 551–571. Michael Lacey and Christoph Thiele, "A proof of boundedness of the Carleson operator", Math. Res. Lett. 7:4 (2000), 361–370. Ole G. Jørsboe and Leif Mejlbro, The Carleson–Hunt theorem on Fourier series.
Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy. Norbert Wiener first proved Wiener's 1/f theorem, [1] see Wiener's theorem. It states that ...
This was disproved by Paul du Bois-Reymond, who showed in 1876 that there is a continuous function whose Fourier series diverges at one point. The almost-everywhere convergence of Fourier series for L 2 functions was postulated by N. N. Luzin , and the problem was known as Luzin's conjecture (up until its proof by Carleson (1966)).
Then the Fourier series of f converges at t to f(t). For example, the theorem holds with ω f = log −2 ( 1 / δ ) but does not hold with log −1 ( 1 / δ ). Theorem (the Dini–Lipschitz test): Assume a function f satisfies = ().
The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.
The convolution of D n (x) with any function f of period 2 π is the nth-degree Fourier series approximation to f, i.e., we have () = () = = ^ (), where ^ = is the k th Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
A version holds for Fourier series as well: if is an integrable function on a bounded interval, then the Fourier coefficients ^ of tend to 0 as . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line.
List of Fourier-related transforms; Fourier transform on finite groups; Fractional Fourier transform; Continuous Fourier transform; Fourier operator; Fourier inversion theorem; Sine and cosine transforms; Parseval's theorem; Paley–Wiener theorem; Projection-slice theorem; Frequency spectrum