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

Tamás Erdélyi¹

¹ Department of Mathematics Texas A&M University College Station, TX 77843, U.S.A.

Studia Mathematica, Tome 155 (2003) no. 2, pp. 145-152

Résumé

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.

Détail

DOI : 10.4064/sm155-2-4

Keywords: denote mathop span ldots collection finite linear combinations functions ldots mathbb principal result paper following theorem full clarkson erd schwartz theorem suppose lambda infty sequence distinct positive numbers mathop span lambda lambda ldots dense only sum infty frac lambda lambda infty moreover sum infty frac lambda lambda infty every function closure mathop span lambda lambda ldots represented analytic function mathbb setminus infty restricted result improves earlier result nbsp borwein erd lyi stating sum infty frac lambda lambda infty every function closure mathop span lambda lambda ldots infty result may viewed improvement extension completion earlier results ntz clarkson erd schwartz borwein erd lyi nbsp johnson operstein