Pointwise Convergence of the Fourier Series

Dirichlet Kernel

Let $f:[-\pi ,\pi ]\to \mathbb{C}$ where

$$S_{N}f=\sum _{k=-N}^N{c}_k{e}^{ikx}{c}_k={\int }_{-\pi }^{\pi }f(x)e^{-ikx}dx$$

Consider the sum

$$\sum _{k=-N}^N{e}^{ikx}=e^{-iNx}+e^{-i(N-1)x}+\cdots +e^{i0x}+\cdots +e^{iNx}$$

By applying the formula for the sum of a geometric series, the sum simplifies to

$$\sum _{k=-N}^N{e}^{ikx}=\frac{\sin (N+\frac{1}{2})x}{\sin (\frac{1}{2}x)}=D_N(x)$$

Significance

The kernel allows us to describe the $N-$th partial sum as a convolution:

$$S_{N}f(x)={\int }_{-\pi }^{\pi }\frac{1}{2\pi }f(t)D_N(x-t)dt\approx f\ast D_N$$

Theorem (Lipschitz Continuous)

If $f(x)$ is Lipschitz Continuous at $x_0\in [-\pi ,\pi ]$, then

$$(\exists M>0)(\exists \delta >0)(|x-x_0|<\delta ⟹|f(x)-f(x_0)|<M|x-x_0|)$$

then ${\lim }_{N\to \infty }{S}_{N}f(x_0)=f(x)$.