Fourier Transforms

Let there be a periodic function, $F(x)$, with a period $X$ such that $\forall$ $x$, $$F(x) = F(x + X)$$ Now, we consider such a real valued, periodic function, $f(x)$, where $x$ is a real variable and take $X = 2\pi$ thus, $$f(x) = f(x + 2\pi)$$ Then a general form of the Fourier series is given by $$f(x) = \frac{a_0}{2} + \sum^{\infty}_{n=1}a_n \cos(nx) + \sum^{\infty}_{n=1}a_n \sin(nx)$$ Now, according to Fourier series expansion, we can expand any given ugly looking function into a function of sines and cosines using the Fourier series expansion, except for the cases in which the function have:

• Infinite number of maximas/minimas,
• Infinite number of discontinuities,
• Divergent functions, and 
• Dirichlet functions

Some important relations
When $m$, $n$ $\in$ $\left( . . . -3, -2, -1, 0, 1, 2, 3, . . .\right)$ $$\frac{1}{\pi}\int^{2\pi}_{0}\cos(mx)\cos(nx)dx = \frac{1}{\pi}\int^{2\pi}_{0}\sin(mx)\sin(nx)dx = \delta_{mn}$$ $$\frac{1}{\pi}\int^{2\pi}_{0}\sin(mx)\cos(nx)dx = 0$$ Now, the coefficients in the Fourier series expansion of $f(x)$ is determined by the analytic expressions, $$a_n = \frac{1}{\pi} \int^{2\pi}_{0} f(x)\cos(nx)dx$$ $$b_n = \frac{1}{\pi} \int^{2\pi}_{0} f(x)\sin(nx)dx$$ N.B. The space of periodic functions = Hilbert space (infinite dimensional).

Now, consider an integer $n$ such that,
\begin{align}
e^{+inx} &= \cos(nx) + i\sin(nx)\\
e^{-inx} &= \cos(nx) - i\sin(nx)
\end{align}

Now, using this information we can write the function as $$f(x) = \sum^{\infty}_{-\infty} C_n e^{inx}$$ Where, $C_0 = \frac{1}{2}a_0$, $C_n = \frac{1}{2}(a_n - i b_n)$ and $C_{-n} = \frac{1}{2}(a_n + i b_n)$; $\forall n > 0$

Here, $e^{inx}$ and $e^{-inx}$ form an orthonormal basis, i.e. $$\frac{1}{2\pi}\int^{2\pi}_{0}e^{inx}e^{-inx}dx = \delta_{mn} = \left<n|m\right>$$ Then, $|f\left> = \sum^{\infty}_{n = -\infty} C_n |n\right>$ where, $|n\left> \equiv e^{inx}\right.$ and $|f\left> \equiv f(x) \right.$ and, $$\left<f|g\right> = \frac{1}{2\pi} \int^{2\pi}_{0}f^{*}(x)g(x)dx$$

Example: 1) Sawtooth wave
Consider a function, $f(x)$ such that $f(x) = f(x + 2\pi)$ and,
\begin{align}
f(x) &= x, &x \in (0,\pi)\\
&= x - 2\pi, &x \in (\pi, 2\pi)
\end{align}\begin{align}
F(x) &= 2\left[\sin(x) - \frac{1}{2}\sin(2x) + \frac{1}{3}\sin(3x) - . . . + \frac{(-1)^{n+1}}{n}\sin(nx) + . . . \right]\\
F(x) &= -F(-x)
\end{align}

Example: 2)
\begin{align} \sum^{\infty}_{n=1}\frac{sin(nx)}{n} &= -\frac{1}{2}(\pi + x) ,  &x \in (-\pi, 0)\\
&= +\frac{1}{2}(\pi - x) ,  &x \in (0, +\pi) \end{align}

Example: 3) Square waves
\begin{align}
\sum^{\infty}_{n=0}\frac{\sin[(2n+1)x]}{(2n+1)} &= -\frac{\pi}{4}, &-\pi < x <0\\
&= \frac{\pi}{4}, &0< x < \pi
\end{align}

Example: 4)
\begin{align}
\sum^{\infty}_{n=1}\frac{cos(nx)}{n} &= -\ln\left[2.\sin\left(\frac{|x|}{2}\right)\right], &x \in (-\pi, \pi)
\end{align}
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