The multiplicative completion of sets of functions
R. P. Boas, Harry Pollard · Bulletin of the American Mathematical Society · 1948
A set {f n (x) }f of functions of L 2 (a, 6), where (a, b) is finite or infinite, is called complete if g(x) ÇJL 2 and Jlf n {x)g{x)dx = 0, w = l, 2, • • • , imply that g(x) = 0 almost everywhere on (a, b); a well known equivalent property ("closure") is that every element of L 2 can be approximated in the L 2 metric by finite linear combinations of the /n(#).Suppose that {f n (x)} is not complete.It will sometimes be possible to find a function m(x) such that the set {m(x)f n (x)} is complete.This can also be considered as completeness after a change of weight function or a change of measure; but we shall not attempt to consider the most general change of measure here.We give some results on when a set can or cannot be completed by multiplication ; the problem of finding necessary and sufficient conditions is left open.We first state our results.