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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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