Editing Convergence of Fourier series (section)

==Preliminaries==

Consider ''f'' an [[Lebesgue integration|integrable]] function on the interval {{closed-closed|0, 2''π''}}. 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;∞.