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There exist continuous functions whose Fourier series converges pointwise but not uniformly. [8] However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L 1 (T) and the Banach–Steinhaus uniform boundedness principle.
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.
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)).
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 ...
Existence or divergence to infinity of the Cesàro mean is also implied. By a theorem of Marcel Riesz, Fejér's theorem holds precisely as stated if the (C, 1) mean σ n is replaced with (C, α) mean of the Fourier series (Zygmund 1968, Theorem III.5.1).
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.
An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30. Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.
The sum of an absolutely convergent Fourier series is continuous, so () where C(T) is the ring of continuous functions on the unit circle. On the other hand an integration by parts, together with the Cauchy–Schwarz inequality and Parseval's formula, shows that