Binary bases of spaces of continuous functions. (Q1417971)

The authors consider the following problem. Let \(\alpha=(\alpha_0, \alpha_1,\dots)\) and set \(S_\alpha=\text{ span} \{t^n-\alpha_nt^{n+1}:n=0,1,\dots\}\), and let \(C[a,b]\) be the Banach space of continuous functions on the finite interval \([a,b]\) with norm defined by \(\| f\| =\max_{t\in[a,b]}| f(t)| \). This paper addresses the question of density of \(S_\alpha\) in \(C[a,b]\). The authors show that this problem is quite complex and, in fact, variants of this problem are equivalent to the general moment problem. Let \(C[a,b]^*\) be the topological dual of \(C[a,b]\). Fix \([a,b]\). The moment problem \(\beta_k=F(t^k)\) for \(k=0,1,\dots\), has a solution \(F\in C[a,b]^*\) if and only if \([\beta_kt^j-\beta_jt^k:0\leq k,j]\neq C[a,b]\). The authors also show that this problem is related to a series of operator questions and they explore the relation of these operator equations to the moment problem.