The ``Full Clarkson–Erdős–Schwartz Theorem'' on the closure of non-dense Müntz spaces Academic Article uri icon

abstract

  • Denote by span{f 1 , f 2 , . . .} the collection of all finite linear combinations of the functions f 1 , f 2 , . . . over ℝ. The principal result of the paper is the following. THEOREM (Full Clarkson-Erdos-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 j2 + 1 = ∞. Moreover, if ∑ j=1∞ λ j /λ j2 + 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 j2 + 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, Erdos, L. Schwartz, P. Borwein, Erdélyi, W. B. Johnson, and Operstein.

author list (cited authors)

  • Erdélyi, T.

citation count

  • 5

publication date

  • January 2003