The Full Müntz Theorem in C [0, 1] and L 1 [0, 1]
Peter Borwein, Tamás Erdélyi · Journal of the London Mathematical Society · 1996
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 infiλi = 0 while supiλi = ∞. The paper settles the L∞ and L1 cases of the following. THEOREM (Full Müntz Theorem in Lp[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 Lp[0, 1] if and only if Σ i = 0 ∞ λ i + 1 / p ( λ i + 1 / p ) 2 + 1 = ∞ .