The Full Mntz Theorem inLp[0, 1] for 0Academic Article
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Denote by span{f1, f2, . . .} the collection of all finite linear combinations of the functions f1, f2, . . . over . The principal result of the paper is the following. Theorem (Full Mntz Theorem in Lp(A) for p (0, ) and for compact sets A [0,1] with positive lower density at 0). Let A [0,1] be a compact set with positive lower density at 0. Let p (0, ). Suppose (j)j=1 is a sequence of distinct real numbers greater than -(1/p). Then span{x1, x2, . . .} is dense in Lp(A) if and only if j=1 j+(1/p)/(j+(1/p))2+1 = . Moreover, if j=1 j+(1/p)/(j+(1/p))2+1 < , then every function from the Lp(A) closure of span{x1, x2, . . .} can be represented as an analytic function on {z (-,0] : |z| < rA} restricted to A (0,rA), where rA := sup{y : m(A [y, )) > 0} (m() denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Mntz, Szsz, Clarkson, Erds, P. Borwein, Erdlyi, and Operstein. Related issues about the denseness of span{x1, x2, . . .} are also considered.
