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

### Tom 155 / 2003

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

#### Streszczenie

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.

#### Autorzy

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

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