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

Tamás Erdélyi. "The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces." Studia Mathematica 155.2 (2003): 145-152. <http://eudml.org/doc/286227>.

@article{TamásErdélyi2003,
abstract = {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.},
author = {Tamás Erdélyi},
journal = {Studia Mathematica},
keywords = {Müntz theorem; Clarkson-Erdős-Schwartz theorem; denseness in },
language = {eng},
number = {2},
pages = {145-152},
title = {The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces},
url = {http://eudml.org/doc/286227},
volume = {155},
year = {2003},
}

TY - JOUR
AU - Tamás Erdélyi
TI - The "Full Clarkson-Erdős-Schwartz Theorem" on the closure of non-dense Müntz spaces
JO - Studia Mathematica
PY - 2003
VL - 155
IS - 2
SP - 145
EP - 152
AB - 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.
LA - eng
KW - Müntz theorem; Clarkson-Erdős-Schwartz theorem; denseness in
UR - http://eudml.org/doc/286227
ER -