Generalizations of Muntz's Theorem via a Remez-type inequality for Muntz spaces
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The principal result of this paper is a Remez-type inequality for Mntz polynomials: formular presented or equivalently for Dirichlet sums: formular presented where 0 = 0 < 1 < 2 < . The most useful form of this inequality states that for every sequence (i)i=0 satisfying i=1 1/i < , there is a constant c depending only on A := (i)i=0and s (and not on n, . or A) so that P for every Mntz polynomial p. as above, associated with (i)i=0x. and for every set .A [Q, 1] of Lebesgue measure at least s > 0. Here || || 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 i=1x 1/i < x, then the set of products {p1p2 : p1 p2 span{x0 , x1 ....}} is not dense in f'O. 1]. A REMEZ-TYPE INEQUALITY The second is a complete extension of Mntz's classical theorem on the denseness of Mntz spaces in C[0, 1] to denseness in C(A). where A [0, ) is an arbitrary compact set with positive Lebesgue measure. That is, for an arbitrary compact set A [0, ) with positive Lebesgue measure, span{x0,x1 ,. . . } is dense in C(A) if and only if i=1 1/i = . Several other interesting consequences are also presented.