#!/usr/bin/env python
# coding: utf-8

# # Gibbs oscillations

# Let us try to approximate a discontinuous function using Chebyshev interpolation
# $$
# f(x) = \begin{cases}
# 1 & x < 0 \\
# 0 & x > 0
# \end{cases}
# $$

# In[40]:


get_ipython().run_line_magic('matplotlib', 'inline')
get_ipython().run_line_magic('config', "InlineBackend.figure_format = 'svg'")
import numpy as np
import matplotlib.pyplot as plt

xmin, xmax = -1.0, +1.0
fun = lambda x: (x < 0)*1.0


# We will interpolate this function at Chebyshev points.

# In[43]:


# Plot function on a finer grid
xx = np.linspace(xmin,xmax,100);
ye = fun(xx);
plt.figure(figsize=(10,8))
plt.plot(xx,ye,'--')
    
for N in range(10,15):
    theta = np.linspace(0,pi,N+1)
    x = np.cos(theta)
    y = fun(x);
    P = np.polyfit(x,y,N);
    yy = np.polyval(P,xx);
    plt.plot(xx,yy)


# The interpolants are oscillatory and this phenomenon is known as Gibbs oscillations.