This paper introduces Fourier duality for a class of affine iterated function systems (IFS) T_i. These systems are determined by a finite family of contractive affine maps in R^d. Our Fourier duality applies to the resulting probability measure mu which is fixed by (T_i). When the IFS is given, the support of the associated mu is a compact set X in R^d, typically a fractal. Our Fourier duality refers to the Hilbert space L^2(X, mu): We show that under a certain unitarity condition involving a pair of affine iterated function systems (T_i) and (S_j) it is possible to recursively construct a Fourier bases in the Hilbert space L^2(X, mu) with the Fourier basis for one depending on the other.
We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these fractals. We prove that as sets these fractals arise as boundaries of functions in closed subspaces of the Hardy space $H^2$. By this we mean that there are lacunary subsets $Γ$ of the non-negative integers, and associated closed $Γ$-subspace in the Hardy space $H^2(\bd)$, $\bd$ denoting the disk, such that for every function $f$ in in $H^2(Γ)$, and for eve…
Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajlasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.
We describe a class of measurable subsets $Ω$ in $\br^d$ such that $L^2(Ω)$ has an orthogonal basis of frequencies $e_λ(x)=e^{i2πλ\cdot x}(x\inΩ)$ indexed by $λ\inΛ\subset\br^d$. We show that such spectral pairs $(Ω,Λ)$ have a self-similarity which may be used to generate associated fractal measures $μ$ with Cantor set support. The Hilbert space $L^2(μ)$ does not have a total set of orthogonal frequencies, but a harmonic analysis of $μ$ may be built instead from a natural representation of the Cuntz C$^*$- algebra which is constructed from a pair of lattices supporting the given spectral pair …
The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.
We consider finite energy and $L^2$ differential forms associated with strongly local regular Dirichlet forms on compact connected topologically one-dimensional spaces. We introduce notions of local exactness and local harmonicity and prove the Hodge decomposition, which in our context says that the orthogonal complement to the space of all exact 1-forms coincides with the closed span of all locally harmonic 1-forms. Then we introduce a related Hodge Laplacian and define a notion harmonicity for finite energy 1-forms. As as corollary, under a certain capacity-separation assumption, we prove th…
Jorgensen and Pedersen have proven that a certain fractal measure $ν$ has no infinite set of complex exponentials which form an orthonormal set in $L^2(ν)$. We prove that any fractal measure $μ$ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in $L^2(μ)$ such that the frequencies have positive Beurling dimension.
We prove a general ampliation homogeneity result for the quasicentral modulus of an n-tuple of operators with respect to the (p,1) Lorentz normed ideal. We use this to prove a formula involving Hausdorff measure for the quasicentral modulus of n-tuples of commuting Hermitian operators the spectrum of which is contained in certain Cantor-like self-similar fractals.
For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra. In this paper we investigate this problem for general fractal measures induced by iterated function systems (IFS). We prove some existence results of spectra associated with Hadamard pairs. We also obtain some characterizations of Bessel spectrum in terms of finite matrices for affine IFS measures, and one sufficient condition of frame spectrum in the case th…
We generalize the compatible tower condition given by Strichartz to the almost-Parseval-frame tower and show that non-trivial examples of almost-Parseval-frame tower exist. By doing so, we demonstrate the first singular fractal measure which has only finitely many mutually orthogonal exponentials (and hence it does not admit any exponential orthonormal bases), but it still admits Fourier frames.
We show that a Krein-Feller operator is naturally associated to a fixed measure $μ$, assumed positive, $σ$-finite, and non-atomic. Dual pairs of operators are introduced, carried by the two Hilbert spaces, $L^{2}\left(μ\right)$ and $L^{2}\left(λ\right)$, where $λ$ denotes Lebesgue measure. An associated operator pair consists of two specific densely defined (unbounded) operators, each one contained in the adjoint of the other. This then yields a rigorous analysis of the corresponding $μ$-Krein-Feller operator as a closable quadratic form. As an application, for a given measure $μ$, including t…
Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel …
We establish an asymptotic formulas for the eigenvalue counting function of the Schrödinger operator $-Δ+V$ for some unbounded potentials $V$ on several types of unbounded fractal spaces. We give sufficient conditions for Bohr's formula to hold on metric measure spaces which admit a cellular decomposition, and then verify these conditions for fractafolds and fractal fields based on nested fractals. In particular, we partially answer a question of Fan, Khandker, and Strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow-up of a Sierpinski gasket.
The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between solenoid measures associated with classical filters in $\mathbb R^d$ and those associated with filters on inflated fractal sets. In particular, it is shown that the classical case produces atomic fiber measures, and as a result supports both suitably defined solenoid MSF wavelets and systems of imprimitivity for the corresponding wavelet representation of the …
We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincare inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.
We study the construction of exponential frames and Riesz sequences for a class of fractal measures on ${\mathbb R}^d$ generated by infinite convolution of discrete measures using the idea of frame towers and Riesz-sequence towers. The exactness and overcompleteness of the constructed exponential frame or Riesz sequence is completely classified in terms of the cardinality at each level of the tower. Using a version of the solution of the Kadison-Singer problem, known as the $R_ε$-conjecture, we show that all these measures contain exponential Riesz sequences of infinite cardinality. Furthermor…
We study orthogonality relations for Fourier frequencies and complex exponentials in Hilbert spaces $L^2(μ)$ with measures $μ$ arising from iterated function systems (IFS). This includes equilibrium measures in complex dynamics. Motivated by applications, we draw parallels between analysis of fractal measures on the one hand, and the geometry of wavelets on the other. We are motivated by spectral theory for commuting partial differential operators and related duality notions. While stated initially for bounded and open regions in $\br^d$, they have since found reformulations in the theory of f…
We show a norm convergence result for the Laplacian on a class of post-critically finite fractals with arbitrary Borel regular probability measure which can be approximated by a sequence of finite-dimensional graph Laplacians with corresponding discrete probability measures. As a consequence other functions of the Laplacians (heat operator, spectral projections etc.) converge as well in operator norm. One also deduces convergence of the spectrum and the eigenfunctions in energy norm.
