Paper 2, Section I, G

Topics in Analysis | Part II, 2014

State Chebyshev's equal ripple criterion.

Let

h(t)=∏ℓ=1n(t−cos⁡(2ℓ−1)π2n)h(t)=\prod_{\ell=1}^{n}\left(t-\cos \frac{(2 \ell-1) \pi}{2 n}\right)

Show that if q(t)=∑j=0najtjq(t)=\sum_{j=0}^{n} a_{j} t^{j} where a0,…,ana_{0}, \ldots, a_{n} are real constants with ∣an∣⩾1\left|a_{n}\right| \geqslant 1, then

sup⁡t∈[−1,1]∣h(t)∣⩽sup⁡t∈[−1,1]∣q(t)∣\sup _{t \in[-1,1]}|h(t)| \leqslant \sup _{t \in[-1,1]}|q(t)|

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