Convergence of Fourier Series - Pointwise Convergence

Pointwise Convergence

There are many known sufficient conditions for the Fourier series of a function to converge at a given point x, for example if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon).

The Dirichlet–Dini Criterion states that: if ƒ is 2π–periodic, locally integrable and satisfies

then (S_nƒ)(x₀) converges to ℓ. This implies that for any function ƒ of any Hölder class α > 0, the Fourier series converges everywhere to ƒ(x).

It is also known that for any periodic function of bounded variation, the Fourier series converges everywhere. See also Dini test.

There exists a continuous function 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.