The ``Full ClarksonErdsSchwartz Theorem'' on the closure of non-dense Mntz spaces
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Denote by span{f 1 , f 2 , . . .} the collection of all finite linear combinations of the functions f 1 , f 2 , . . . over . The principal result of the paper is the following. THEOREM (Full Clarkson-Erdos-Schwartz Theorem). Suppose ( j ) j=1 is a sequence of distinct positive numbers. Then span{1, x 1 , x 2 , . . .} is dense in C[0, 1] if and only if j=1 j / j2 + 1 = . Moreover, if j=1 j / j2 + 1 = , then every function from the C[0, 1] closure of span{1, x 1 , x 2 , . . .} can be represented as an analytic function on {z (-, 0]: |z| < 1} restricted to (0, 1). This result improves an earlier result by P. Borwein and Erdlyi stating that if j=1 j / j2 + 1 = , then every function from the C[0, 1] closure of span{1, x 1 , x 2 , . . .} is in C (0, 1). Our result may also be viewed as an improvement, extension, or completion of earlier results by Mntz, Szsz, Clarkson, Erdos, L. Schwartz, P. Borwein, Erdlyi, W. B. Johnson, and Operstein.
