Convergence of Fourier Series - Preliminaries

Preliminaries

Consider ƒ an integrable function on the interval . For such an ƒ the Fourier coefficients 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. In order to investigate this more carefully, the partial sums need to be defined:

The question we will be interested in is: do the functions (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.

Before continuing the Dirichlet kernel needs to be introduced. Taking the formula for, inserting it into the formula for and doing some algebra will give 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 will play a crucial role in the discussion. The norm of D_n in L1(T) coincides with the norm of the convolution operator with D_n, acting on the space C(T) of periodic continuous functions, or with the norm of the linear functional ƒ → (S_nƒ)(0) on C(T). Hence, this family of linear functionals on C(T) is unbounded, when n → ∞.