On subwavelet sets

In this note we give a characterization of subwavelet sets and show that any point x E R\O has a neighborhood which is contained in a regularized wavelet set. In [1] the notion of a wavelet set was introduced and in [8] subwavelet sets were considered. Wavelet sets were also introduced independently and simultaneously as the support sets of MSF (Minimally Supported FIrequency) wavelets in the sequence of papers [3], [5], and [6]. (See also the recent excellent book [4].) The purpose of this note is to provide a characterization of the subwavelet sets and to use this characterization to prove that every point x E R\{O} has a neighborhood contained in a regularized wavelet set. (Regularized wavelet sets are wavelet sets with certain nice properties; see [7].) In particular, this shows that the union of the interiors of all wavelet sets is R\{0}. We begin by introducing some preliminary terminology and notation. The measure space under consideration will always be R together with its a-ring L of Lebesgue measurable subsets and Lebesgue measure ,u. Recall (cf. [1]) that a function w E L2 (R):= L2(R, L, ,) is a wavelet if the family of (equivalence classes of) functions {Wj,k}j,kez defined by Wj,k(S) = 2 12w(2is + k), s E iR, j, k E Z, is an orthonormal basis for L 2(R). A subset G of R with positive measure is a wavelet set if 1 XG = F(w), where w is a wavelet in L2(R) and F is the Fourier-Plancherel transform on L 2(R). A measurable subset G of R is called a regularized wavelet set if the family {G + 2k7r}kEz is a partition of R and the family {2kG}kez is a partition of R\{O}. For two measurable subsets F and G of iR, we write F G if ,u(F V G) = 0. It is proved in [7] that if W is any wavelet set, then there exists a regularized wavelet set W' such that W' W. A measurable subset G of R is translation congruent modulo 27r to a (measurable) set H c R if there exists a measurable bijection : G (G) such that p(s) s is an integral multiple of 27r for every s in G and W(G) H. Analogously, G c R\{0} is said to be dilation congruent modulo 2 to a (measurable) set H if there exists a measurable bijection : G -+ V(G) such that V)(s)/s is an integral power of 2 for every s in G and +)(G) H. Let r: R -* E :[-27r, -7r) U [7r, 27r) be the function defined by r(x) x + 2jir, where j is the unique integer satisfying x + 2jir E E, and let 6: R\{0}-* E be the function defined by 6(x) = 2kx, where k is the unique integer Received by the editors April 11, 1997. 1991 Mathematics Subject Classification. Primary 42C15. 3549 ( 1998 American Mathematical Society This content downloaded from 157.55.39.144 on Wed, 07 Sep 2016 05:43:40 UTC All use subject to http://about.jstor.org/terms 3550 EUGEN J. IONASCU AND CARL M. PEARCY for which 2kx E E. For a function f: X -+ X and k E Z we write f(?) for the map x -> x on X and f(k) for the composition of f [resp. f'] with itself IkI times if k > 0 [resp. k E defined by hi :=IGi o (TIG,)-l and h2 := TIG2 o (61G2)-l It is clear that hi and h2 are measurable injections. We now construct a new map h from h, and h2 following the idea of the proof of the Cantor-Bernstein theorem in set theory. To increase the clarity of the presentation we write E := E and consider h1 E -E and h2: E E. We denote f = h2 o h: E E and g := h1 o h2: E, and note that these maps are measurable injections by Remark 1. One can check that E and E can be partitioned as follows: E = EOCJ (UEkUE)k kEN E = io00 U Ekk E)k -kEN where Eo =n f () (E), = n 9( (E) jEN jEN Ek f(k)(E)\(f (k-)o h2)(E), El (f(k-I)oh2)(E)\f( )(E), k E N,