The full Muntz theorem in C[0,1] and L(1)[0,1]

The main result of this paper is the establishment of the 'full Muntz Theorem' in C[0, 1]. This characterizes the sequences {lambda(i)}(infinity)(i-1) of distinct, positive real numbers for which span 1, x(lambda 1), x(lambda 2),...) is dense in C[0, 1]. The novelty of this result is the treatment of the most difficult case when inf(i) lambda(i) = 0 while sup(i) lambda(i) = infinity. The paper settles the L(infinity) and L(i) cases of the following. THEOREM (Full Muntz Theorem in L(p)[0,1]). Let p is an element of[1,infinity]. Suppose that {lambda(i)}(i)(i-0)nfinity is a sequence of distinct real numbers greater than -1/p. Then span {x(lambda 0),x lambda 1,...} is dense in L?p[0,1] if and only if [GRAPHICS] d only if [GRAPHICS] +...+ X(n)</SUP>; moreover, we show that the sequence {C-n} is asymptotic to (2/pi)e(1-y)(2n)(n). These results generalize work of the first author on binary forms and will likely find application in the enumeration of solutions of decomposable form inequalities.