The “Full Clarkson–Erdős–Schwartz Theorem” on the closure of non-dense Müntz spaces
Volume 155 / 2003
Abstract
Denote by $\mathop{\rm span} \{f_1, f_2, \ldots\}$ the collection of all finite linear combinations of the functions $f_1, f_2, \ldots$ over ${\mathbb R}$. The principal result of the paper is the following.
Theorem (Full Clarkson–Erdős–Schwartz Theorem). Suppose $(\lambda_j)_{j=1}^\infty$ is a sequence of distinct positive numbers. Then $\mathop{\rm span} \{1, x^{\lambda_1}, x^{\lambda_2}, \ldots\}$ is dense in $C[0,1]$ if and only if $$ \sum^{\infty}_{j=1} \frac{\lambda_j}{\lambda_j^2 + 1} = \infty . $$ Moreover, if $$ \sum_{j=1}^{\infty} \frac{\lambda_j}{\lambda_j^2+1} < \infty , $$ then every function from the $C[0,1]$ closure of $\mathop{\rm span} \{1, x^{\lambda_1}, x^{\lambda_2}, \ldots\}$ can be represented as an analytic function on $\{z \in {\mathbb C} \setminus (-\infty, 0]: |z| < 1\}$ restricted to $(0,1)$.
This result improves an earlier result by P. Borwein and Erdélyi stating that if $$ \sum_{j=1}^{\infty} \frac{\lambda_j}{\lambda_j^2+1} < \infty , $$ then every function from the $C[0,1]$ closure of $\mathop{\rm span} \{1, x^{\lambda_1}, x^{\lambda_2}, \ldots\}$ is in $C^\infty(0,1)$. Our result may also be viewed as an improvement, extension, or completion of earlier results by Müntz, Szász, Clarkson, Erdős, L. Schwartz, P. Borwein, Erdélyi, W. B. Johnson, and Operstein.
