Convergence of Fourier series

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, L^p spaces, summability methods and the Cesàro mean.

Consider ƒ an integrable function on the interval [0,2π]. For such an ƒ the Fourier coefficients ${\displaystyle {\widehat {f}}(n)}$ are defined by the formula

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

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

The question here is: Do the functions ${\displaystyle S_{N}(f)}$ (which are functions of the variable t we omitted in the notation) converge to ƒ and in which sense? Are there conditions on ƒ ensuring this or that type of convergence? This is the main problem discussed in this article.

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