Chebyshev interpolation

Definition

interpolation on top of Chebyshev node

Node

The selection of node can be:

• Chebyshev–Gauss: $x_j=\cos (\pi \frac{2j+1}{2n})$
• Chebyshev–Lobatto
• etc

Implementation

it can be implemented by Lagrange Interpolation, orthogonal polynomial interpolation, and etc.
OR

import numpy as np
import numpy.linalg as la
 
n = len(y)
x = np.cos((2 * np.array(list(range(n))) + 1)*np.pi/(2*n))
V = np.zeros((n,n))
V[:,0] = 1
V[:,1] = x
for i in range(2, n):
    V[:,i] = 2*x*V[:,i-1] - V[:,i-2]
 
c = la.solve(V,y)

Crude Error

\max |f(x) - p_{n-1}(x)| \approx \frac{C}{2^{n-1}}(b-a) ^n$$ where - $C$ constant - $n$ number of nodes - $b-a$ the range