Fourier Series¶

Background¶

Cos Method Fourier Series

Educational script demonstrating cos method fourier series concepts.

Code¶

```python """ Cos Method Fourier Series

Educational script demonstrating cos method fourier series concepts. """

@title Fourier Series Of Non-Periodic Function on \([-\pi,\pi]\)¶

f = lambda x : np.sin( ( x - 1.8 ) ** 2 )

======================================================================¶

def main(): n = 10_000 theta = np.linspace(-np.pi,np.pi,n) d_theta = theta[1] - theta[0] f_theta = f(theta)

deg = 100
f_recovered = np.zeros_like(f_theta)
for k in range(deg):
    A_k = np.sum( f_theta[:-1] * np.cos(k*theta[:-1]) ) * d_theta / np.pi
    B_k = np.sum( f_theta[:-1] * np.sin(k*theta[:-1]) ) * d_theta / np.pi
    if k == 0:
        A_k /= 2
        B_k /= 2
    f_recovered += A_k * np.cos(k*theta) + B_k * np.sin(k*theta)

fig, ax = plt.subplots(1,1,figsize=(12,4))
ax.plot(theta,f_theta,label='Original',lw=10,alpha=0.3)
ax.plot(theta,f_recovered,"--r",label=f'Recovered with {n} Terms')
ax.legend()
plt.show()

if name == "main": main() ```

Exercises¶

Exercise 1. For \(f(x) = \sin((x - 1.8)^2)\) on \([-\pi, \pi]\), explain why the Fourier series includes both cosine and sine terms.

Solution to Exercise 1

The function \(f(x) = \sin((x-1.8)^2)\) is neither even nor odd about \(x = 0\) (it has no symmetry). Therefore both Fourier coefficients \(A_k\) and \(B_k\) are nonzero. The cosine terms capture the even part \(\frac{f(x) + f(-x)}{2}\) and the sine terms capture the odd part \(\frac{f(x) - f(-x)}{2}\).

Exercise 2. The Fourier coefficients are computed as \(A_k = \frac{1}{\pi}\int_{-\pi}^{\pi} f(\theta)\cos(k\theta)d\theta\). In the code, this integral is approximated by a Riemann sum. What is the order of this approximation?

Solution to Exercise 2

With \(n = 10{,}000\) points and spacing \(d\theta = 2\pi/n\), the left-endpoint Riemann sum has error \(O(d\theta) = O(1/n)\). For \(n = 10{,}000\), the integration error per coefficient is \(\sim 10^{-4}\). Using the trapezoidal rule (averaging left and right endpoints) would give \(O(1/n^2) \sim 10^{-8}\) accuracy.

Exercise 3. If \(f\) has a discontinuity on \([-\pi, \pi]\), the Fourier series exhibits the Gibbs phenomenon. Describe what happens near the discontinuity as \(N \to \infty\).

Solution to Exercise 3

Near the discontinuity, the partial Fourier sum overshoots by approximately 9% of the jump size, regardless of \(N\). As \(N\) increases, the overshoot becomes narrower (width \(\sim \pi/N\)) but its height remains \(\sim 0.089 \times \text{jump}\). The Fourier series converges pointwise everywhere except at the discontinuity, but not uniformly. This affects the COS method for digital options, where the payoff has a jump.

Exercise 4. For the smooth function \(f(x) = \sin((x-1.8)^2)\) with \(\deg = 100\) terms, the recovery is visually perfect. Estimate the convergence rate for smooth periodic functions.

Solution to Exercise 4

For infinitely differentiable periodic functions, the Fourier coefficients decay faster than any polynomial: \(|A_k|, |B_k| = O(k^{-m})\) for all \(m > 0\). The truncation error with \(N\) terms is \(O(N^{-m})\) for all \(m\), giving superalgebraic convergence. In practice, for analytic functions the convergence is exponential: \(O(e^{-\alpha N})\). With 100 terms, the error is below machine precision for this smooth function.