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Concept
Convergence of Fourier series
Résumé
In mathematics, the question of whether the Fourier series of a periodic function converges to a 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.
Consider f an integrable function on the interval . For such an f the Fourier coefficients are defined by the formula
It is common to describe the connection between f 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 of whether a Fourier series converges is: Do the functions (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 , inserting it into the formula for and doing some algebra gives that
where ∗ stands for the periodic convolution and is the Dirichlet kernel, which has an explicit formula,
The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely
a fact that plays a crucial role in the discussion. The norm of Dn in L1(T) coincides with the norm of the convolution operator with Dn,
acting on the space C(T) of periodic continuous functions, or with the norm of the linear functional f → (Snf)(0) on C(T). Hence, this family of linear functionals on C(T) is unbounded, when n → ∞.
In applications, it is often useful to know the size of the Fourier coefficient.
If is an absolutely continuous function,
for a constant that only depends on .
Source officielle
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