MARKOV- AND BERNSTEIN-TYPE INEQUALITIES FOR M ¨ UNTZ POLYNOMIALS AND EXPONENTIAL SUMS IN L p Tam´ as Erd´ elyi Abstract. The principal result of this paper is the following Markov-type inequality for M¨ untz polynomials. Theorem (Newman’s Inequality in Lp[a, b] for [a, b] ⊂ (0, ∞)). Let Λ := (λ j ) ∞ j=0 be an increasing sequence of nonnegative real numbers. Suppose λ 0 =0 and there exists a δ> 0 so that λ j ≥ 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 λ j 1 A ‖P ‖ Lp[a,b] for every P ∈ Mn(Λ), where Mn(Λ) denotes the linear span of {x λ 0 ,x λ 1 ,...,x λn } 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. Erd´ elyi [3] for 1 ≤ p ≤∞. Note that the interval [0, 1] plays a special role in the study of M¨ untz 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. 1. Introduction and Notation Let P n denote the collection of all algebraic polynomials of degree at most n with real coefficients. For notational convenience let ‖·‖ [a,b] := ‖·‖ L∞[a,b] . The following two inequalities, together with their various extensions, play an important role in approximation theory. See, for example, DeVore and Lorentz [8], Lorentz [10], and Natanson [12]. Theorem 1.1 (Markov’s Inequality). If p ∈P n , then ‖p ′ ‖ [−1,1] ≤ n 2 ‖p‖ [−1,1] . 1991 Mathematics Subject Classification. Primary: 41A17, Secondary: 30B10, 26D15. Key words and phrases. M¨ untz polynomials, lacunary polynomials, exponential sums, Dirichlet sums, Markov-type inequality, Bernstein-type inequality. Research is supported, in part, by NSF under Grant No. DMS–9623156. Typeset by A M S-T E X 1