Fourier Series

Fourier Coefficients

For $f\in L^1(\mathbb{T})$ define the Fourier coefficients by $\hat{f}(n):={\int }_{\mathbb{T}}f(x)e^{-2\pi inx}dx,n\in \mathbb{Z}.$

$N$-th partial sum of the Fourier series

For integers $N\ge 0$ define the $N$-th partial sum of the Fourier series by $S_{N}f(x)={\sum }_{n=-N}^N\hat{f}(n)e^{2\pi inx}.$

Proposition

For every $f\in L^1(\mathbb{T})$, we have $S_{N}f\in C(\mathbb{T}).$ and $S_{N}f(x)=f\ast D_N(x),$ where $D_N$ is the $N$-th Dirichlet kernel $D_N(x):={\sum }_{n=-N}^N{e}^{2\pi inx}=\frac{\sin ((2N+1)\pi x)}{\sin (\pi x)}$ and convolutions are defined by $f\ast g(x)={\int }_{\mathbb{T}}f(x-y)g(y)dy={\int }_{\mathbb{T}}f(y)g(x-y)dy.$

Consider the space $L^2(S^1)$, the set $\{e^{in\theta }{\}}_{n\in \mathbb{Z}}$ is an orthonormal basis, and for each function $f\in L^2(S^1)$, the corresponding $n$th coefficient is the $n$th Fourier coefficient: $a_n(f)=\frac{1}{2\pi }{\int }_{\pi }^{\pi }f(x)e^{-inx}\text{dd}x.$This can be seen by the following two facts:

• The Parseval’s identity holds for $L^2(S^1)$: ${\sum }_{n=-\infty }^{\infty }|a_n(f){|}^2=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }|f(x){|}^2\text{dd}x.$
• The Fourier series converges to $f$ in the $L^2$ norm: ${\lim }_{N\to 0}\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{\left|f(x)-{\sum }_{n\le N}{a}_n{e}^{inx}\right|}^2\text{dd}x=0$ Indeed, the mapping $f\mapsto \{a_n(f){\}}_{n\in \mathbb{Z}}$ makes $L^2(S^1)$ (unitarily) “equivalent” to ${\ell }^2(\mathbb{Z})$, we shall seriously define this equivalence, and prove this in the next section. (See the proposition.)