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IMC 1995 Problem 10

Source: IMC 1995

February 19, 2021
functionapproximationreal analysis

Problem Statement

a) Prove that for every ϵ>0\epsilon>0 there is a positive integer nn and real numbers λ1,…,λn\lambda_{1},\dots,\lambda_{n} such that max⁡x∈[−1,1]∣x−∑k=1nλkx2k+1∣<ϵ.\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon. b) Prove that for every odd continuous function ff on [−1,1][-1,1] and for every ϵ>0\epsilon>0 there is a positive integer nn and real numbers μ1,…,μn\mu_{1},\dots,\mu_{n} such that max⁡x∈[−1,1]∣f(x)−∑k=1nμkx2k+1∣<ϵ.\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.
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