Chebyshev Polynomial

$T_n(x)=\cos (n{\cos }^{-1}(x))$ is an orthogonal sequence of polynomials satisfying:

• $T_1(x)=1$
• $T_2(x)=x$
• $T_{n+1}=2x{T}_n-T_{n-1}$, so all terms are polinomials
• ${⟨T_n,T_m⟩}_{[-1,1]}={\delta }_{nm}\sqrt{1-n^2}$, where
  • ${\delta }_{nm}$ is the Kronecker delta
  • ${⟨\cdot ,\cdot ⟩}_{[-1,1]}$ is the L2 norm over the interval $[-1,1]$.

$T_n$ attains the $\pm 1$ bounds in the $[-1,1]$ interval:

Info

The above can be neatly used for building low resource approximations of any function, as described in the blog post Chebyshev Approximation and How It Can Help You Save Money, Win Friends, and Influence People.