We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension. Moreover, the spectrum is then typically quasi-periodic, but non-periodic, i.e., the spectrum is a ``small perturbation'' of a lattice. Due to earlier research on IFSs, there are known results on certain classes of spectral duality-pairs, also called spectral pairs or spectral measures. It is known that some duality pairs are associated with complex Hadamard matrices. However, not all IFSs $X$ admit spectral duality. When $X$ is given, we identify geometric conditions on $X$ for the existence of a Fourier spectrum, serving as the second part in a spectral pair. We show how these spectral pairs compose, and we characterize the decompositions in terms of atoms. The decompositions refer to tensor product factorizations for associated complex Hadamard matrices.
The idea of expanding L 2 -functions on subsets Ω in Euclidean space into bases of more fundamental functions is central, and dates back to Fourier. It is of use in signal processing and in physics, but also of interest in its own right: Here we are thinking of Fourier series, orthogonal polynomials, eigenfunctions for Hamiltonians in physics, and wavelets; to mention only a few. These are instances of Marc Kac's question: "Can you hear the shape of a drum?" Each case suggests a natural choice of basis functions. We will consider a setting when the set Ω under consideration comes with some degree of selfsimilarity, and we will be asking for the possibility of choosing Fourier bases; i.e., we will examine the possibility of selecting orthonormal bases in L 2 (Ω, µ) where µ is a finite measure on Ω which reflects the intrinsic selfsimilarity. Such selfsimilarity arises for example in affine iterated function systems [Hut81], but it is much more general as we demonstrate. If Ω has non-empty interior, it is natural to take µ to be the restriction of Lebesgue measure. Hence we are faced with a pair of subsets in R d : (1) the set Ω itself, and (2) the points λ which make up the frequencies in some candidate for a Fourier basis in L 2 (Ω).
In the discussion below, we recall the history of the problem, and earlier results by a number of authors which are relevant for our present work. The central question we address here is this: “To what extent may some given set Ω in d dimensions be built up from atoms of fundamental blocks in such a way that the spectral data for the “atoms” determine that of Ω itself?” Even if the spectral data for the atoms is periodic, we show that for composite systems, the expectation is quasiperiodicity in a sense we make precise in section 3 below.
Our work is inspired by [Fug74,IKT01,Lon67,Lab02] among others. We consider open subsets Ω in R d of finite positive Lebesgue measure. Our focus is on the case when the Hilbert space L 2 (Ω) has an orthogonal Fourier basis, i.e., an orthogonal basis complex exponentials. The measure on Ω is taken to be the restriction of d-dimensional Lebesgue measure. The exponents in such an orthogonal basis will then form a discrete subset Λ in R d . We say that (Ω, Λ) is a spectral pair and Ω is a spectral set.
We identify a geometric condition which characterizes spectral pairs arising as attractors of iterated function systems (IFSs), i.e., from a finite set of affine mappings in R d .
We analyze sets of the form A + [0, 1] where A is some finite set of integers, and find conditions when such a set is spectral (Theorem 3.25). We characterize those sets which are attractors of an affine IFS (Theorem 3.27 and show that they are spectral sets (Theorem 3.29). We construct a new class of spectral measures (Theorem 3.21), and obtain a counterexample to a conjecture of Laba and Wang (Example 3.9). We present an example of a measure which has an infinite family of mutually orthogonal exponentials but is not spectral (Proposition 3.23). We show how new spectra can be constructed from old for some fractal measures (Lemma 3.33 and Theorem 3.35). We construct a connected spectral domain in R 3 which does not tile R 3 by any lattice (Example 4.3).
We introduce more general spectral pairs than the (Ω, Λ) systems, including a pairing for finite subsets in R d , and from IFSs. And we introduce an operation on spectral pairs. Our idea is to identify an interplay between finite spectral pairs on the one hand, and a class of infinite Euclidean ones on the other, those built on affine iterated function system (IFS) measures, see Definition 2.4. With tools from IFS-theory, this then allows us to exploit our new results on finite systems in extending some of the classical constructions from Fuglede’s paper [Fug74].
Section 3 contains several new results: (a) A FFT-type algorithm (Corollary 3.19) in 1D of building molecules of spectral pairs (Ω, Λ) from atoms. (b) For this class of spectral pairs (Ω, Λ), when Ω is fixed, we find all the possible sets Λ which serve as spectra (Theorem 3.25.) In section 4 we consider systems in higher dimensions, with special attention to the case when Ω is both open and connected.
The broader motivation for our paper is a set of intriguing connections between tiles, spectrum and wavelet analysis. To a large degree, the role of scaling operators has been missing in many early approaches to spectral-tile duality. The advent of wavelets [Dau92] did much to remedy this. Some early papers stressing the role played by scaling and selfsimilarity in spectrum-tile duality and in wavelets are [Law91, BJ99, JP99, BJR99], and especially [GM92] which make useful connections to signal processing in engineering. Our main results concern spectral properties implied by selfsimilarity.
The implications of this selfsimilarity (i.e., similarity up to a suitable scaling operation, or a group of affine mappings) take several forms: Our Corollary 3.4 below id
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