A REMEZ - TYPE INEQUALITY 3 Theorem 1

The principal result of this paper is a Remez-type inequality for M untz polynomials: p(x) := n X i=0 a i x i ; or equivalently for Dirichlet sums: P(t) := n X i=0 a i e ? i t ; where (i) 1 i=0 is a sequence of distinct real numbers. The most useful form of this inequality states that for every sequence (i) 1 i=0 satisfying 1 X i=0 i 6 =0 1 j i j < 1 there is a constant c depending only on (i) 1 i=0 , A, , and (and not on n or A) so that the inequality kpk ;;] c kpk A holds for every M untz polynomial p, as above, associated with (i) 1 i=0 , for every set A 0; 1) of positive Lebessgue measure, and for every ; ] (ess inf A; ess sup A): Here k k A denotes the supremum norm on A. This Remez-type inequality allows us to resolve several problems. Most notably we show that the M untz-type theorems of Clarkson, Erd} os, and Schwartz on the denseness of spanfx 0 ; x 1 ; :: : g; i 2 R distinct on a; b], a > 0, remain valid with a; b] replaced by an arbitrary compact set A (0; 1) of positive Lebesgue measure. This extends earlier results of the authors under the assumption that the numbers i are nonnegative.