Korovkin approximations in L_p-spaces

William George Kitto and Daniel Eliot Wulbert

Vol. 63 (1976), No. 1, 153–167

DOI: 10.2140/pjm.1976.63.153

Abstract

The main result is a characterization of finite Korovkin sets for positive operators in l_p. It follows that a finite set containing a positive function is a Korovkin set in l_p if and only if it is a Korovkin set in c₀. The methods also show:

Proposition. Let X be a compact subset of Rⁿ. Let K be a subspace of C(X) containing the constants. If K is a Korovkin set in C(X), then K is Korovkin set in L_p(X).

Several related results are also given. For example a question of G. G. Lorentz about the restrictions of Korovkin set in C(X) to a subset Y ⊆ X is answered.

Mathematical Subject Classification 2000

Primary: 41A65

Milestones

Received: 11 July 1973

Revised: 15 July 1974

Published: 1 March 1976