The “Full Clarkson–Erdős–Schwartz Theorem” on the closure of non-dense Müntz spaces

Volume 155 / 2003

Tamás Erdélyi Studia Mathematica 155 (2003), 145-152 MSC: Primary 30B60, 41A17. DOI: 10.4064/sm155-2-4

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.

Authors

  • Tamás ErdélyiDepartment of Mathematics
    Texas A&M University
    College Station, TX 77843, U.S.A.
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image