MARKOV- AND BERNSTEIN-TYPE INEQUALITIES FOR MÜNTZ POLYNOMIALS AND EXPONENTIAL SUMS IN Lp

The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem (Newman’s Inequality in Lp[a, b] for [a, b] ⊂ (0,∞)). Let Λ := (λj) ∞ j=0 be an increasing sequence of nonnegative real numbers. Suppose λ0 = 0 and there exists a δ > 0 so that λj ≥ δj for each j. Suppose 0 < a < b and 1 ≤ p ≤ ∞. Then there exists a constant c(a, b, δ) depending only on a, b, and δ so that ‖P ‖Lp[a,b] ≤ c(a, b, δ) 0 @ n X j=0 λj 1 A ‖P‖Lp[a,b] for every P ∈ Mn(Λ), where Mn(Λ) denotes the linear span of {xλ0 , xλ1 , . . . , xλn} over R. When p = ∞ this has been shown in [5]. When [a, b] = [0, 1] and with ‖P ‖Lp [a,b] replaced with ‖xP (x)‖Lp [a,b] this was proved by D. Newman [13] for p = ∞ and by P. Borwein and T. Erdélyi [3] for 1 ≤ p ≤ ∞. Note that the interval [0, 1] plays a special role in the study of Müntz spaces Mn(Λ). A linear transformation y = αx+β does not preserve membership in Mn(Λ) in general (unless β = 0). So the analogue of Newman’s Inequality on [a, b] for a > 0 does not seem to be obtainable in any straightforward fashion from the [0, b] case.