Markov-Type Inequalities for Products of Müntz Polynomials

Abstract

Let Λ≔(λj)∞j=0 be a sequence of distinct real numbers. The span of {xλ0, xλ1, …, xλn} over R is denoted by Mn(Λ)≔span{xλ0, xλ1, …, xλn}. Elements of Mn(Λ) are called Müntz polynomials. The principal result of this paper is the following Markov-type inequality for products of Müntz polynomials. Theorem2.1.LetΛ≔(λj)∞j=0andΓ≔(γj)∞j=0be increasing sequences of nonnegative real numbers. LetK(Mn(Λ), Mm(Γ))≔sup‖x(pq)′(x)‖[0, 1]‖pq‖[0, 1]:p∈Mn(Λ),q∈Mm(Γ).Then13((m+1)λn+(n+1)γm)⩽K(Mn(Λ), Mm(Γ))⩽18(n+m+1)(λn+γm).In particular ,23(n+1)λn⩽K(Mn(Λ), Mn(Λ))⩽36(2n+1)λn. Under some necessary extra assumptions, an analog of the above Markov-type inequality is extended to the cases when the factor x is dropped, and when the interval [0, 1] is replaced by [a, b]⊂(0, ∞).