Fourier Theorem

Definition

Suppose that

• $f\in C_p(-\pi ,\pi )$^[Piecewise Continuous]
• $f$ in $2\pi$-periodic on $\mathbb{R}$
• For some $-\infty <x<\infty$, $f_{+}^{\prime }(x)$ and $f_{-}^{\prime }(x)$ both exist

Then, the Fourier series $\frac{a_0}{2}+\sum _{n=1}^{\infty }(a_n\cos (nx)+b_n\sin (nx))$, $a_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (nx)dx$, $b_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx$ converges to $\frac{1}{2}(f(x^{+})+f(x^{-}))$

Facts

Bessel’s Inequality for Fourier Series

Let $f\in C_p(-\pi ,\pi )$[^1], then $f$ satisfies Bessel’s inequality
$A_N=\frac{a_0^2}{2}+\sum _{n=1}^N(a_n^2+b_n^2)\le \frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x{)}^{2}dx$

Since $\{A_n\}$ is increasing bounded sequence of real number, $\{A_n\}$ converges. $\Rightarrow \underset{n\to \infty }{\lim }{a}_n=\underset{n\to \infty }{\lim }{b}_n=0$

where $a_0=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)dx,a_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\cos (nx)dx,b_n=\frac{1}{\pi }{\int }_{-\pi }^{\pi }f(x)\sin (nx)dx$

Two Lemmas

Let $G(u)$ be Piecewise Continuous on $(0,\pi )$ Then, $\underset{N\to \infty }{\lim }{\int }_0^{\pi }\sin (Nu+\frac{u}{2})G(u)du=0$

Let $g(u)$ be Piecewise Continuous on $(0,\pi )$ and $g_{+}^{\prime }(0)$ Then, ${\int }_0^{\pi }{D}_N(u)g(u)du\to \frac{\pi }{2}g(0^{+})$ as $N\to \infty$ where $D_N$ is Dirichlet Kernel