, Volume 76, Issue 1-2, pp 183-192

Notes on lacunary Müntz polynomials

Abstract

We prove that a Müntz system has Chebyshev polynomials on [0,1] with uniformly bounded coefficients if and only if it is lacunary. A sharp Bernstein-type inequality for lacunary Müntz systems is established as well. As an application we show that a lacunary Müntz system fails to be dense inC(A) in the uniform norm for everyA ⊂ [0,1] with positive outer Lebesgue measure. A bounded Remez-type inequality is conjectured for non-dense Müntz systems on [0,1] which would solve Newman’s problem concerning the density of products of Müntz systems.