chebyshev equioscillation theorem

[[concept]]

Idea

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 $h(\lambda )={\sum }_{k=0}^{K^{\prime }-1}{h}_k{T}_k(\lambda )$ ie $K^{\prime }<K$ for the same approximation quality.

Proof (intuition only)

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

link:: wikipedia

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