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Miklós Schweitzer 1984- Problem 7

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September 5, 2016
college contestsalgebrapolynomial

Problem Statement

7. Let VV be a finite-dimensional subspace of C[0,1]C[0,1] such that every nonzero f∈Vf\in V attains positive value at some point. Prove that there exists a polynomial PP that is strictly positive on [0,1][0,1] and orthogonal to VV, that is, for every f∈Vf \in V,
∫01f(x)P(x)dx=0\int_{0}^{1} f(x)P(x)dx =0 (F.39) [A. Pinkus, V. Totik]
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