In mathematics, the question of whether the Fourier series of a given periodic function converges to the given function is studied in classical harmonic analysis, a branch of pure mathematics. Convergence does not occur in the general casecertain criteria must be met.

Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, L<sup>p</sup> spaces, summability methods and the Cesàro mean.

Preliminaries

Consider f an integrable function on the interval . For such an f the Fourier coefficients <math>\widehat{f}(n)</math> are defined by the formula

:<math>\widehat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}f(t)e^{-int}\,\mathrm{d}t, \quad n \in \Z.</math>

It is common to describe the connection between f and its Fourier series by

:<math>f \sim \sum_n \widehat{f}(n) e^{int}.</math>

The notation ~ here means that the sum represents the function in some sense. To investigate this more carefully, the partial sums must be defined:

:<math>S_N(f;t) = \sum_{n=-N}^N \widehat{f}(n) e^{int}.</math>

The question of whether a Fourier series converges is: Do the functions <math>S_N(f)</math> (which are functions of the variable t we omitted in the notation) converge to f and in which sense? Are there conditions on f ensuring this or that type of convergence?

Before continuing, the Dirichlet kernel must be introduced. Taking the formula for <math>\widehat{f}(n)</math>, inserting it into the formula for <math>S_N</math> and doing some algebra gives that

:<math>S_N(f)=f * D_N</math>

where ∗ stands for the periodic convolution and <math>D_N</math> is the Dirichlet kernel, which has an explicit formula,

:<math>D_n(t)=\frac{\sin((n+\frac{1}{2})t)}{\sin(t/2)}.</math>

The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely

:<math>\int |D_n(t)|\,\mathrm{d}t \to \infty </math>

a fact that plays a crucial role in the discussion. The norm of D<sub>n</sub> in L<sup>1</sup>(T) coincides with the norm of the convolution operator with D<sub>n</sub>,

acting on the space C(T) of periodic continuous functions, or with the norm of the linear functional f&nbsp;→ (S<sub>n</sub>f)(0) on C(T). Hence, this family of linear functionals on C(T) is unbounded, when n&nbsp;→&nbsp;∞.

Magnitude of Fourier coefficients

In applications, it is often useful to know the size of the Fourier coefficient.

If <math>f</math> is an absolutely continuous function,

:<math>\left|\widehat f(n)\right|\le {K \over |n|}</math>

for <math>K</math> a constant that only depends on <math>f</math>.

If <math>f</math> is a bounded variation function,

:<math>\left|\widehat f(n)\right|\le