GENERALIZATIONS OF MUNTZS THEOREM VIA A REMEZ-TYPE INEQUALITY FOR MUNTZ SPACES

The principal result of this paper. is a Remez-type inequality for Mun tz polynomials: p(x):= (n) Sigma(i=0) aix(lambda i), or equivalently f or Dirichlet sums: P(t) := (n) Sigma(i=0) aie(-lambda it), where 0 = l ambda(0) < lambda(1) < lambda(2) < .... The most useful form of this i nequality states that for every sequence (lambda(i))(infinity)(i=0) sa tisfying Sigma(i=1)(infinity) 1/lambda(i) < infinity, there is a const ant c depending only on Lambda := (lambda(i))(infinity)(i=0) and s (an d not on n, q, or A) so that parallel to p parallel to([0,q]) less tha n 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 ever y set A subset of [q, 1] of Lebesgue measure at least s > 0. Here para llel to .parallel to(A) denotes the supremum norm on A. This Remez-typ e inequality allows us to resolve two reasonably long-standing conject ures. The first conjecture it lets us resolve is due to D. J. Newman a nd 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 ele ment of span {x(lambda 0), x(lambda 1),...}} is not dense in C[0,1]. T he 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 sub set of [0, infinity) is an arbitrary compact set with positive Lebesgu e measure. That is, for an arbitrary compact set A subset of [0, infin ity) 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) = i nfinity. Several other interesting consequences are also presented.