Generalizations of Muntz's Theorem via a Remez-type inequality for Muntz spaces

The principal result of this paper. is a Remez-type inequality for Muntz polynomials: p(x):= (n) Sigma(i=0) aix(lambda i), or equivalently for Dirichlet sums: P(t) := (n) Sigma(i=0) aie(-lambda it), where 0 = lambda(0) < lambda(1) < lambda(2) < .... The most useful form of this inequality states that for every sequence (lambda(i))(infinity)(i=0) satisfying Sigma(i=1)(infinity) 1/lambda(i) < infinity, there is a constant c depending only on Lambda := (lambda(i))(infinity)(i=0) and s (and not on n, q, or A) so that parallel to p parallel to([0,q]) less than or equal to c parallel to p parallel to A for every Muntz polynomial p, as above, associated with (lambda(i))(infinity)(i=0), and for every set A subset of [q, 1] of Lebesgue measure at least s > 0. Here parallel to .parallel to(A) denotes the supremum norm on A. This Remez-type inequality allows us to resolve two reasonably long-standing conjectures. The first conjecture it lets us resolve is due to D. J. Newman and dates from 1978. It asserts that if Sigma(i=1)(infinity) 1/lambda(i) < infinity, then the ser of products {p(1)p(2) : P-1, p(2) is an element of span {x(lambda 0), x(lambda 1),...}} is not dense in C[0,1]. The second is a complete extension of Muntz's classical theorem on the denseness of Muntz spaces in C[0, 1] to denseness in C(A), where A subset of [0, infinity) is an arbitrary compact set with positive Lebesgue measure. That is, for an arbitrary compact set A subset of [0, infinity) with positive Lebesgue measure, span{x(lambda 1), x(lambda 1),...} is dense in C(A) if and only if Sigma(i=1)(infinity) 1/lambda(i) = infinity. Several other interesting consequences are also presented.