A Spanning Set for C(I n )

Abstract

F(In ) denotes the Banach space of continuous functions on the unit n-cube, In, in Rn . Let {a'}, i = 0, 1, 2, . be a countable colii lection of n-tuples of positive real numbers satisfying liml aJ = + oo for j = 1, . n. We canonically enlarge the family of monomials {Xa } to a family of functions Sr(A) . Conjecture. The linear span of 9(A) is dense in '(In) if and only if Z`0 /lla1l = +oo. For n = 1 this is equivalent to the Muntz-Szasz theorem. For n > 1 we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants G, N > 1 such that aiI 0) be dense in the space of continuous functions on the unit n-cube In = [0, 1 ]n. An essentially equivalent problem is the question of characterizing discrete sets of uniqueness for functions of several variables bounded in a cone. Interesting results on this pro-blem have been found by Korevaar-Hellerstein [7], Ronkin [11, 12], and Berndtsson [1]. For example, in [1] the following result is proved (see also [1 1, 12]). Received by the editors November 11, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 32E30; Secondary 41A63.