Fourier Series

A Fourier series decomposes a periodic function into a sum of sines and cosines. For a function f(x) with period 2L, the series is:

$f_N(x) = \frac{a_0}{2} + \sum_{n=1}^{N} \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right]$

The Fourier coefficients are computed via integration:

• DC component: $a_0 = \frac{1}{L} \int_{-L}^{L} f(x)\, dx$
• Cosine coefficients: $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx$
• Sine coefficients: $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx$

Numerical Integration

We represent f(x) by N evenly spaced sample points on [-L, L) using a midpoint grid:

$x_i = -L + \frac{2L(i + 0.5)}{N}, \quad i = 0, 1, \ldots, N-1$

The step size is $\Delta x = 2L/N$. The integrals become sums:

$a_0 \approx \frac{1}{L} \sum_{i=0}^{N-1} f(x_i)\, \Delta x$

$a_n \approx \frac{1}{L} \sum_{i=0}^{N-1} f(x_i) \cos\left(\frac{n\pi x_i}{L}\right) \Delta x$

$b_n \approx \frac{1}{L} \sum_{i=0}^{N-1} f(x_i) \sin\left(\frac{n\pi x_i}{L}\right) \Delta x$

Square Wave Example

A square wave on $[-\pi, \pi]$ is $f(x) = 1$ if $x > 0$, else $-1$. Its Fourier series contains only odd sine harmonics:

$f(x) = \frac{4}{\pi} \left[ \sin(x) + \frac{\sin(3x)}{3} + \frac{\sin(5x)}{5} + \cdots \right]$

So $b_1 = 4/\pi \approx 1.2732$, $b_3 = 4/(3\pi) \approx 0.4244$, and all $a_n = 0$.

Your Task

Implement the four Fourier series functions. Use N=1000 sample points on the midpoint grid. The f_values input is a list of N function values at the grid points.