2008/08/21 by Dorin Ervin Dutkay, Dutkay, Dorin Ervin, Palle E. T. Jørgensen +2
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #advanced mathematical theories #math.FA
paper · pdf · doi:10.48550/arxiv.0808.2946
openalex publication_date 2008/08/21 · arxiv created 2008/11/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let d be a positive integer, and let μ be a finite measure on \brd. In this paper we ask when it is possible to find a subset Λ in \brd such that the corresponding complex exponential functions eλ indexed by Λ are orthogonal and total in L2(μ). If this happens, we say that (μ, Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L2(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in \brd; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.