page_title: Convergence of Fourier series
section_title: Pointwise convergence
hierarchical_section_title: Convergence of Fourier series / Pointwise convergence
caption_attribution_description: English: Superposition of Fourier components of a sawtooth wave
context_page_description: In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.
  Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lᵖ spaces, summability methods and the Cesàro mean.
context_section_description: 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 converges 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 (Sₙƒ)(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.
  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
caption: Superposition of sinusoidal wave basis functions (bottom) to form a sawtooth wave (top); the basis functions have wavelengths λ/k (k=integer) shorter than the wavelength λ of the sawtooth itself (except for k=1). All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the Gibbs phenomenon
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