On interpolation families of wavelet sets

The question of which groups are isomorphic to groups of interpolation maps for interpolation families of wavelet sets was raised by Dai and Larson. In this article it is shown that any finite group is isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. A dyadic orthonormal (or orthogonal) wavelet is a function 4 E L2 (R) (Lebesgue measure), with the property that the set {2"0'(2't 1): n, l E 2} is an orthonormal basis for L2 (R). For certain measurable sets E C R the normalized characteristic function 72==7rXE is the Fourier transform of such a wavelet. There are several characterizations of such sets. In [DL] they are called wavelet sets, and the corresponding wavelet is called an s-elementary wavelet. In [FW],[HWW1],[HWW2] they are the support sets of MFS (minimal frequency supported) wavelets. In [DL], a method of operator-theoretic interpolation between certain special pairs of wavelets, and between single wavelets and special families of them is developed. The interpolation pairs of wavelet sets are extensively discussed and the method yields a new construction of a class of wavelets due to Meyer. An interpolation pair is the simplest case of an interpolation family (see definitions below). Affiliated to an interpolation family is a group (under composition) of measure preserving transformations of R called the interpolation maps. Problem E in [DL] asked for a characterization of all groups which are isomorphic to a group of interpolation maps for some interpolation family of wavelet sets. They showed, by example, that the cyclic groups 22 and 23 of order 2 and 3 could be realized in this way, but they proved no cases of other groups. The purpose of this paper is to prove that every finite group can be realized in this way. Before introducing the characterization of wavelet sets, we will need some more terminology. In [DL], two measurable sets E, F are called translation congruent modulo 2wr if there is a measurable bijection 0: E -> F such that s(s) s is an integral multiple of 2wr for each s E E. Analogously, two measurable sets G, H are called dilation congruent modulo 2 if there is a measurable bijection T: G -> H such that for each s E G there is an integer n depending on s, such that T (s) = 2s. We note in passing that if Ei and Fi are measurable subsets of R such that Ei and Received by the editors July 7, 1998 and, in revised form, November 24, 1998. 2000 Mathematics Subject Classification. Primary 42C40.