\[\cos(x)=\frac{1}{2}(e^{ix}+e^{-ix})\] \[\sin(x)=\frac{1}{2i}(e^{ix}-e^{-ix})\] \[e^{ix}=\cos(x)+i\sin(x)\] \[|e^{ix}|^2=e^{ix}\overline{e^{ix}}=e^{ix}e^{-ix}=1\] then \[|e^{inx}|=1\] Let \(x_0\) be the smallest positive number such that \(\cos(x_0)=0\), we define the number \(\pi\) by \(\pi=2x_0\). Then \(\cos(\pi/2)=0\) Because \(|\cos(\pi/2)+i\sin(\pi/2)|=\sqrt{\cos^2(\pi/2)+\sin^2(\pi/2)}=1\) then \(\sin^2(\pi/2)=1\) Since \(\sin'(x)=\cos(x)>0\) in \((0,\pi/2)\), \(\sin(x)\) is increasing in \((0,\pi/2)\), hence \(\sin(\pi/2)=1\). Thus \[e^{\frac{\pi}{2}i}=\cos(\pi/2)+i\sin(\pi/2)=i\] \[e^{\pi i}=\cos(\pi)+i\sin(\pi)=-1\] \[e^{-\pi i}=\cos(-\pi)+i\sin(-\pi)=-1\] \[e^{2\pi i}=\cos(2\pi)+i\sin(2\pi)=1\] \[e^{z+2\pi i}=\cos(z+2\pi)+i\sin(z+2\pi)=\cos(z)+i\sin(z)=e^z\quad(\text{z complex})\] Then \(e^{ix}\) is periodic, with period \(2\pi i\). \[\int_{-\pi}^{\pi} e^{inx}dx=\frac{e^{inx}}{in}\Biggl|_{-\pi}^{\pi}=\begin{cases} 2\pi & (\text{if } n=0) \\ 0 & (\text{if } n=\pm1,\pm2,\cdots) \end{cases}\]
The trigonometric polynomial is \[\begin{align} f(x)&=a_0+\sum_{n=1}^{N}(a_n\cos nx+b_n\sin nx)\\ &=a_0+\sum_{n=1}^{N}\Biggl[(a_n\frac{1}{2}(e^{inx}+e^{-inx})+b_n\frac{1}{2i}(e^{inx}-e^{-inx}))\Biggr]\\ &=\sum_{-N}^{N}c_ne^{inx}\quad(x\text{ is real}, a_0,\cdots,a_N,b_1,\cdots,b_N \text{ are complex}) \end{align}\] When \(N\to\infty\), \[\sum_{-\infty}^{\infty}c_ne^{inx}\quad(x\text{ is real})\] is the trigonometric series. If \(n\) is a nonzero integer, \(e^{inx}\) has period \(2\pi\), hence \[\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{inx}dx=\begin{cases} 1 & (\text{if } n=0) \\ 0 & (\text{if } n=\pm1,\pm2,\cdots) \end{cases}\] Let us multiply \[f(x)=\sum_{-N}^{N}c_ne^{inx}\] by \(e^{-imx}\), where \(m\) is an integer, and integrate the product \[c_m=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-imx}dx\quad(|m|\leq N)\]
Let \(\{\phi_n\}\quad(n=1,2,3,\cdots)\) be a sequence of complex functions on \([a,b]\), such that \[\int_{a}^{b}\phi_n(x)\overline{\phi_m(x)}dx=0\quad(n\ne m)\] Then \(\{\phi_n\}\) is said to be an orthogonal system of functions on \([a,b]\). If \[\int_{a}^{b}|\phi_n(x)|^2dx=1\] for all \(n\), \(\{\phi_n\}\) is said to be orthonormal. Because \(|e^{inx}|=1\) \[\int_{-\pi}^{\pi}\Bigl|\frac{1}{\sqrt{2\pi}}e^{inx}\Bigr|^2dx=\frac{1}{2\pi}\int_{-\pi}^{\pi}|e^{inx}|^2dx=1\] so the functions \(\frac{1}{\sqrt{2\pi}}e^{inx}\) form an orthonormal system on \([-\pi,\pi]\).
If \(\{\phi_n\}\) is orthonormal on \([a,b]\) and if \[c_n=\int_{a}^{b}f(t)\overline{\phi_n(t)}dt\quad(n=1,2,3,\cdots)\] we call \(c_n\) the \(n^{th}\) Fourier coefficient of \(f\) relative to \(\{\phi_n\}\). We call the series \[f(x)\sim\sum_{1}^{\infty}c_n\phi_n(x)=\sum_{1}^{\infty}\Biggl(\int_{a}^{b}f(t)\overline{\phi_n(t)}dt\Biggr)\phi_n(x)\] the Fourier series of \(f\) relative to \(\{\phi_n\}\).
Let \(\{\phi_n\}\) be orthonormal on \([a,b]\). Let \[s_n(x)=\sum_{m=1}^{n}c_m\phi_m(x)\] be the \(n^{th}\) partial sum of the Fourier series of \(f\), and suppose \[t_n(x)=\sum_{m=1}^{n}\gamma_m\phi_m(x)\] Then \[\int_{a}^{b}|f-s_n|^2dx\leq\int_{a}^{b}|f-t_n|^2dx\] and equality holds if and only if \[\gamma_m=c_m\quad(m=1,\cdots,n)\] That is to say \(s_n\) gives the best possible mean square approximation to \(f\). Let \(\int\) denote the integral over \([a,b]\), \(\sum\) the sum from \(1\) to \(n\). Then \[\int f\overline{t}_n=\int f\sum\overline{\gamma}_m\overline{\phi}_m=\sum c_m\overline{\gamma}_m\] by the definition of \(\{c_m\}\), since \(\{\phi_m\}\) is orthonormal, so \[\int|t_n|^2=\int t_n\overline{t}_n=\int\sum\gamma_m\phi_m\sum\overline{\gamma}_k\overline{\phi}_k=\sum|\gamma_m|^2\] \[\begin{align} \int|f-t_n|^2&=\int|f|^2-\int f\overline{t}_n-\int\overline{f}t_n+\int|t_n|^2\\ &=\int|f|^2-\sum c_m\overline{\gamma}_m-\sum \overline{c}_m\gamma_m+\sum\gamma_m\overline{\gamma}_m\\ &=\int|f|^2+\sum (\gamma_m-c_m)\overline{\gamma}_m-\sum \overline{c}_m\gamma_m+\sum \overline{c}_mc_m-\sum \overline{c}_mc_m\\ &=\int|f|^2+\sum (\gamma_m-c_m)\overline{\gamma}_m-\sum (\gamma_m-c_m)\overline{c}_m-\sum \overline{c}_mc_m\\ &=\int|f|^2+\sum (\gamma_m-c_m)(\overline{\gamma}_m-\overline{c}_m)-\sum \overline{c}_mc_m\\ &=\int|f|^2-\sum|c_m|^2+\sum|\gamma_m-c_m|^2 \end{align}\] which is evidently minimized if and only if \[\gamma_m=c_m\quad(m=1,\cdots,n)\] when \(\gamma_m=c_m\), \[\int_{a}^{b}|f(x)|^2dx\ge \sum_{m=1}^{n}|c_m|^2=\int_{a}^{b}|s_n(x)|^2dx\] Since \(\int|f-t_n|^2\ge 0\). When \(n\to\infty\) \[\int_{a}^{b}|f(x)|^2dx\ge \sum_{n=1}^{\infty}|c_n|^2\]
We consider functions \(f\) that have period \(2\pi\) and that are Riemann-integrable on \([-\pi,\pi]\). The Fourier series of \(f\) is then \[\sum_{-\infty}^{\infty}c_ne^{inx}\quad(x\text{ is real})\] whose coefficients \(c_n\) are given by the integrals \[c_m=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-imx}dx\] and \[s_N(x)=s_N(f;x)=\sum_{-N}^{N}c_ne^{inx}\] is the Nth partial sum of the Fourier series of \(f\). Then \[\frac{1}{2\pi}\int_{-\pi}^{\pi}|s_N(x)|^2dx=\sum_{-N}^{N}|c_n|^2\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^2dx\] We define the Dirichlet kernel as \[D_N(x)=\sum_{n=-N}^{N}e^{inx}\] then \[\begin{align} s_N(x)&=\sum_{-N}^{N}\Biggl[\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt\Biggr] e^{inx}\\ &=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\sum_{-N}^{N}e^{in(x-t)}dt\\ &=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)D_N(x-t)dt\\ \end{align}\]