Closure theorems for some discrete subgroups of 𝑅^{𝑘}

Abstract

Introduction. The study of lattice-point translates of an L2 function was begun in [2] where it was required that the set of these functions form an orthogonal set. In this paper, the study is continued, but the orthogonality condition is dropped. Thus, given a function K in L2 on Rk, we should like to know when its latticepoint translates are dense in the largest subspace of L2 for which this is possible, i.e. the subspace of L2 functions F such that the support of F, the Fourier transform of F, is contained in the support of K?. The problem is solved in Theorem 1, and the solution involves a geometric and measure-theoretic condition on S, the support of ?. Some more or less immediate corollaries follow which clarify the situation. It is the subgroup of lattice-points and certain linear images of it to which the title of the paper refers in this context. In the second theorem however, we consider, for the first and last occasion in this paper, a closure result for certain nondiscrete subgroups to which the already established methods are applicable. In the second section, results analogous to the preceding are established for the case LP, 1 <p < oo. They are, as to be expected, less precise than for the L2 case. In the last section, the main theorem is established in the setting of locally compact, abelian groups. In the matter of notation, m, n,... will stand for lattice-points. Thus, m= (ml, M2,.. ., Mk) where each mj is an integer. LP(S) will denote the space of LP functions F such that the support of F, written as supp F, is in the set S. The set S is normally associated with an L2 function K such that supp K= S. As usual, Xs is the characteristic function of S. S2, will denote the hypercube of side 27T, center at the origin, and with sides parallel to the coordinate axes. It is thus identified with the k torus. H*(K) signifies the L2 closure of the linear span of the functions K(x+ m). Hp*(K) has an analogous meaning for LP closure. 1. Density of translates. For the statement of our first theorem, a class of measurable sets which will serve as supports for acceptable functions KI must be identified. We say that S is of special form if, for almost every x,