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

• Autorzy: Tamás Erdélyi
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 30B60: Completeness problems, closure of a system of functions
• 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol\cprime ski{\u\i}-type inequalities)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 155
• Numer: 2
• Strony: 145-152

Daty

wydano

2003

Twórcy

autor

Tamás Erdélyi

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

Identyfikatory

DOI

10.4064/sm155-2-4