then the Fourier series converges to the average of the left and right limits (but see Gibbs phenomenon). The Dirichlet–Dini Criterion states that: if ƒ is 2–periodic, locally integrable and satisfies then (Snf)(x0) converges to ℓ. This implies that for any function f of any Hölder class α > 0, the Fourier series converges everywhere to f(x). It is also known that for any periodic function of bounded variation, the Fourier series converges everywhere. See also Dini test. In general, the most common criteria for pointwise convergence of a periodic function f are as follows: If f satisfies a Holder condition, then its Fourier series converges uniformly. If f is of bounded variation, then its Fourier series converges everywhere. If f is continuous and its Fourier coefficients are absolutely summable, then the Fourier series converges uniformly. There exist continuous functions whose Fourier series converges pointwise but not uniformly; see Antoni Zygmund, Trigonometric Series, vol. 1, Chapter 8, Theorem 1.13, p. 300. 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 L1(T) and the Banach–Steinhaus uniform boundedness principle. As typical for existence arguments invoking the Baire category theorem, this proof is nonconstructive. It shows that the family of continuous functions whose Fourier series converges at a given x is of first Baire category, in the Banach space of continuous functions on the circle. So in some sense pointwise convergence is atypical, and for most continuous functions the Fourier series does not converge at a given point. However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere. It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by Uniform convergence Suppose , and has modulus of continuity ; then the partial sums of the Fourier series converge to the function with speed for a constant that does not depend upon , nor , nor . This theorem, first proved by D Jackson, tells, for example, that if satisfies the -Hölder condition, then If is periodic and absolutely continuous on , then the Fourier series of converges uniformly, but not necessarily absolutely, to . Absolute convergence A function ƒ has an absolutely converging Fourier series if Obviously, if this condition holds