ISSN:
1432-0940

Keywords:
Primary 28A75, 42B10, 46L55
;
Secondary 05B45
;
Iterated function system
;
Affine maps
;
Fractional measure
;
Harmonic analysis
;
Hilbert space
;
Operator algebras

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics

Notes:
Abstract We consider affine systems inR n constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ bx:=R–1x+b; the corresponding measure μ onR n is a probability measure fixed by the self-similarity $$\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} } $$ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR n such that the exponentials {eiλ·x}Λ form anorthogonal basis forL 2(μ); and (ii) the existence of a certaindual pair of representations of theC *-algebraO N wheren is the cardinality of the setB. (For eachN, theC *-algebraO 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 inL 2(μ) fail to span a dense subspace. Instead we show that theC *-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC *-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) aunitary generalized Hadamard matrix, and (b) a certainsystem 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.

Type of Medium:
Electronic Resource

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