The Full Müntz Theorem in C [0, 1] and L 1 [0, 1]

The main result of this paper is the establishment of the ‘full Müntz Theorem’ in C [0, l]. This characterizes the sequences{ λ i }i = 1 ∞of distinct, positive real numbers for which span{ 1 , x λ 1, x λ 2, … }is dense in C [0, 1]. The novelty of this result is the treatment of the most difficult case when inf i λ i = 0 while sup i λ i = ∞. The paper settles the L ∞ and L 1 cases of the following. THEOREM (Full Müntz Theorem in L p [0,1]). Let p ∈ [l, ∞]. Suppose that{ λ i } i = 0 ∞is a sequence of distinct real numbers greater than −1/ p . Then span{ x λ 0, x λ 1, … }is dense in L p [0, 1] if and only ifΣ i = 0 ∞λ i + 1 / p( λ i + 1 / p ) 2 + 1 = ∞ .