SOLUTION OF BERNSTEIN'S APPROXIMATION PROBLEM'

In his famous monograph on approximation theory [2], S. Bernstein initiated the study of the closure properties of sets of functions {unK(u) }I on the real line. It is supposed that K(u) is continuous on (, o ) and that unK(u) vanishes at u = ? o for each value of n. The problem is to decide when the set { unK(u) } is fundamental in the space C0 of functions continuous on (oo, oo), vanishing at + o, and normed by Ijjfj = max Jf(u) J. So far no necessary and sufficient conditions have been given. A recent paper of Carleson [3 ] reviews most of the known results, but the paper [1 ] which seems to come closest to the true conditions has been overlooked. It is the purpose of this note to give a complete solution. It applies to either realor complex-valued functions and may be read either way.