The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces

Tamás Erdélyi

Studia Mathematica, Tome 157 (2003), p. 145-152 / Harvested from The Polish Digital Mathematics Library

Résumé

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose $(λ_{j})_{j=1}^{∞}$ is a sequence of distinct positive numbers. Then $span{1,x^{λ₁},x^{λ₂},...}$ is dense in C[0,1] if and only if $∑^{∞}_{j=1} (λ_{j})/(λ_{j}²+1) = ∞$. Moreover, if $∑_{j=1}^{∞} (λ_{j})/(λ_{j}²+1) < ∞$, then every function from the C[0,1] closure of $span{1,x^{λ₁},x^{λ₂},...}$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an earlier result by P. Borwein and Erdélyi stating that if $∑_{j=1}^{∞} (λ_{j})/(λ_{j}²+1) < ∞$, then every function from the C[0,1] closure of $span{1,x^{λ₁},x^{λ₂},...}$ is in $C^{∞}(0,1)$. 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

Publié le : 2003-01-01

Zbl 1016.30003

EUDML-ID : urn:eudml:doc:286227