A spanning set for ${\scr C}(I^ n)$

Abstract:

$\mathcal {C}({I^n})$ denotes the Banach space of continuous functions on the unit $n$-cube, ${I^n}$, in ${{\mathbf {R}}^n}$. Let $\{ {a^i}\}$, $i = 0,1,2, \ldots ,$, be a countable collection of $n$-tuples of positive real numbers satisfying ${\operatorname {lim}_i}a_j^i = + \infty$ for $j = 1, \ldots ,n$. We canonically enlarge the family of monomials $\{ {x^{{a^i}}}\}$ to a family of functions $\mathcal {F}(A)$. Conjecture. The linear span of $\mathcal {F}(A)$ is dense in $\mathcal {C}({I^n})$ if and only if $\Sigma _{i = 0}^\infty 1/\left | {{a^i}} \right | = + \infty$. For $n = 1$ this is equivalent to the Müntz-Szasz theorem. For $n > 1$ we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants $G$, $N > 1$ such that $\left | {{a^i}} \right | \leq G{\operatorname {exp}}({i^N})$ for all $i$.