Newman's Inequality for Müntz Polynomials on Positive Intervals

AbstractThe principal result of this paper is the following Markov-type inequality for Müntz polynomials.Theorem(Newman's Inequality on [a, b]⊂(0, ∞)).Let Λ:=(λj)∞j=0be an increasing sequence of nonnegative real numbers. Suppose λ0=0and there exists a δ>0so that λj⩾δj for each j. Suppose0<a<b. Then there exists a constant c(a, b, δ)depending only on a, b, and δ so that[formula]for every P∈Mn(Λ), where Mn(Λ)denotes the linear span of{xλ0, xλ1, …, xλn}overR. When [a, b]=[0, 1] and with ‖P′‖[a, b]replaced with ‖xP′(x)‖[a, b]this was proved by Newman. Note that the interval [0, 1] plays a special role in the study of Müntz spacesMn(Λ). A linear transformationy=αx+βdoes not preserve membership inMn(Λ) in general (unlessβ=0). So the analogue of Newman's Inequality on [a, b] fora>0 does not seem to be obtainable in any straightforward fashion from the [0, b] cas