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Uniform approximation by polynomials...

Source: Miklos Schweitzer, 2014, problem 8.

December 13, 2014
algebrapolynomialfunctioninequalitiesintegrationcalculusderivative

Problem Statement

Let n≥1n\ge 1 be a fixed integer. Calculate the distance inf⁡p,f max⁡0≤x≤1∣f(x)−p(x)∣\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)| , where pp runs over polynomials of degree less than nn with real coefficients and ff runs over functions f(x)=∑k=n∞ckxkf(x)= \sum_{k=n}^{\infty} c_k x^k defined on the closed interval [0,1][0,1] , where ck≥0c_k \ge 0 and ∑k=n∞ck=1\sum_{k=n}^{\infty} c_k=1.
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