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Studia Mathematica

2003 | 155 | 2 | 145-152
Tytuł artykułu

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

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Denote by span{f₁,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.
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Tom
Numer
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145-152
Opis fizyczny
Daty
wydano
2003
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autor
• Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A.
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