chebyshev equioscillation theorem

In conventional signal processing, filters based on Chebyshev polynomials have better cutoff behavior in the spectral domain (compared to general polynomial filters)

• ie, they need fewer coefficients than the Taylor approximation for the same quality

Using $h(\lambda )=\sum _{k=0}^{K-1}{h}_k{\lambda }^k$ requires more coefficients than

Chebyshev Polynomial expansion minimizes $L_{\mathrm{\infty }}$ norm over the approximation interval ($||x|{|}_{\mathrm{\infty }}=\underset{i}{max}(x_i)$), while Taylor series is a local approximation around some $\tilde{\lambda }$