A Density Theorem for Walsh Functions
J. J. Price · Proceedings of the American Mathematical Society · 1967
j. j. price 1.Introduction.A theorem of Levinson [3, p. 13] on the closure of sets of exponentials contains the following result.Let 5= {«,} be an increasing sequence of positive integers.Definewhere A(w) is the number of elements of .S less than n.Suppose D(S) = 1.Then for each positive e, there is a subset E, of the unit interval whose measure exceeds 1-e and such that the family [exp(27TWyx): n -_S\ is total in L2(Et).The proof oi Levinson's theorem requires deep complex methods.In this paper, we obtain an analogous result for Walsh functions using elementary methods.Our result requires the weaker hypothesis that p(5) = l where p(S) = lim sup lim sup (A(» + ft) -A(w))/ft.k-*» n-+»