a plot of a sum of sine waves, emulating the reconstruction from a fourier transfer

plt.py

import numpy as np
import matplotlib.pyplot as plt
# Parameters
PI = np.pi
N_TERMS = 10 # Number of sine terms
NUM_POINTS = 1000 # Number of x values
# Function to compute Fourier sine coefficient a_n for x^3
def compute_coefficient(n):
    x = np.linspace(0, PI, NUM_POINTS)
    y = x**3 * np.sin(n * x)
    h = x[1] - x[0]
    integral = h * (y[0]/2 + np.sum(y[1:-1]) + y[-1]/2)
    return (2 / PI) * integral
# Approximate x^3 using sine series
def approximate_x3(x, coeffs):
    sum = np.zeros_like(x)
    for n in range(1, N_TERMS + 1):
      sum += coeffs[n - 1] * np.sin(n * x)
    return sum
# Compute all coefficients
coeffs = [compute_coefficient(n) for n in range(1, N_TERMS + 1)]
# Evaluate the actual and approximated function
x_vals = np.linspace(-PI, PI, NUM_POINTS)
actual_vals = x_vals**3
approx_vals = approximate_x3(x_vals, coeffs)
# Plot
plt.figure(figsize=(10, 6))
plt.plot(x_vals, actual_vals, label='Actual $x^3$', linewidth=2)
plt.plot(x_vals, approx_vals, label=f'Sine Approx. ({N_TERMS} terms)', linestyle='--')
plt.title('Approximation of $x^3$ using Sine Functions')
plt.xlabel('x')
plt.ylabel('y')
plt.grid(True)
plt.legend()
plt.tight_layout()
plt.show()