On some classical problems concerning L∞-extremal polynomials with constraints
Abstract
First we consider the following problem which dates back to Chebyshev, Zolotarev and Achieser: among all trigonometric polynomials with given leading coefficients a0,...,al, b0,...,bl ∈ R find that one with least maximum norm on [0,2 π]. We show that the minimal polynomial is on [0,2 π] asymptotically equal to a Blaschke product times a constant where the constant is the greatest singular value of the Hankel matrix associated with the τj = aj + i bj. As a special case corresponding statements for algebraic polynomials follow. Finally the minimal norm of certain linear functionals on the space of trigonometric polynomials is determined. As a consequence a conjecture by Clenshaw from the sixties on the behavior of the ratio of the truncated Fourier series and the minimum deviation is proved.