ar X iv : m at h / 06 04 08 7 v 1 [ m at h . FA ] 4 A pr 2 00 6 HARMONIC ANALYSIS OF FRACTAL MEASURES Palle

b∈B μ ◦σ b . There are two a priori candidates for an associated orthogonal harmonic analysis : (i) the existence of some subset Λ in R such that the exponentials {e}λ∈Λ form an orthogonal basis for L (μ); and (ii) the existence of a certain dual pair of representations of the C-algebra ON where N is the cardinality of the set B. (For each N , the C-algebra ON 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(μ) 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, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based on R and B, such that μ has the C-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R,B) satisfies some specific 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 identified.