Uniform approximation by polynomials with integer coefficients

Let r, n be positive integers with n ≥ 6r. Let P be a polynomial of degree at most n on [0,1] with real coefficients, such that [formula] are integers for k = 0,…, r — 1. It is proved that there is a polynomial Q of degree at most n with integer coefficients such that [formula] for x ∈ [0,1], where C1, C2 are some numerical constants. The result is the best possible up to the constants.

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Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.