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absolutely convergent Ak(f algebraic integer array of nodes Bernstein Bernstein's inequality best approximation canonical representation Chebyshev condition Chebyshev expansion Chebyshev polynomials commutes with x2 complex numbers continuous function contradiction convergent convex denned denote distinct points equality holds Equation Erdos established exists extremal for F f1 dx finite fundamental polynomials given hence Hermite interpolation Hint implies integer coefficients interpolating polynomial interval irreducible over Q leading coefficient Lebesgue constants Lemma linear functional mathematical induction minimal polynomial monic monotone increasing nonnegative norm notation nth root obtain orthogonal polynomials partial sums polynomial interpolation polynomial of degree positive primitive nth root Proof properties proved quadrature formula recall Exercise Remark result right-hand side Rolle's theorem root of unity satisfies the Chebyshev sequence Show solution subset Suppose Theorem 2.1 theory trigonometric polynomial unique extremal view of Exercise yields