Harmonic Analysis of Fractal Measures

We consider aane systems in R n constructed from a given integral invertible and expansive matrix R, and a nite set B of translates, b x := R ?1 x + b; the corresponding measure on R n is a probability measure and xed by the selfsimilarity = jBj ?1 P b2B ?1 b. There are two a priori candidates for an associated orthogonal harmonic analysis : (i) the existence of some subset in R n such that the exponentials fe iix g 2 form an orthogonal basis for L 2 (); and (ii) the existence of a certain dual pair of representations of the C-algebra O N where N is the cardinality of the set B. (For each N, the C-algebra O N is known to be simple; it is also called the Cuntz-algebra.) We show that, in the \typical" fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials in L 2 () fail to span a dense subspace. Instead we show that the C-algebraic version of an orthogonal harmonic analysis, viz., (ii), is a natural substitute. It turns out that this version is still based on exponentials e iix , but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of aane systems, based on R and B, such that has the C-algebra property (ii). Speciically, we show that has an orthogonal harmonic analysis (in the sense (ii)) if the system (R; B) satisses some speciic symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) a unitary generalized Hadamard-matrix, and (b) a certain system of lattices which must exist and, at the same time, be compatible with the Hadamard-matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identiied.