Fourier Frames Research Articles

Multi-windowed vertex-frequency analysis for signals on undirected graphs

The recent emerging graph signal processing technologies have been widely applied to analyze signals defined on irregular domains, e.g., data collected from social networks, sensor networks, or transportation systems. Vertex frequency analysis, especially the windowed graph Fourier transform, is one of the most important tools for graph signal analysis and representations. Nevertheless, with a selected window function, it is rather challenging to construct tight frames via the windowed graph Fourier transform. To facilitate the construction of tight frames, in this paper, we consider multi-windowed graph Fourier transforms to develop novel vertex frequency analysis methods. Firstly, under the multi-windowed setting, tight graph Fourier frames are elaborately constructed to fulfill technical demands in different application scenarios. The canonical dual frames of the multi-windowed graph Fourier frames are investigated to establish the reconstruction formulas of graph signals. Additionally, we propose shift multi-windowed graph Fourier frames by directly using the shift operators, e.g., the adjacency matrix. The related tight frames, dual frames and their constructions are also discussed. Experimental results show that the proposed two types of frames can efficiently extract vertex-frequency features of synthetic graph signals. Furthermore, anomaly data can also be detected by these frames.

Fourier Frames for Surface-Carried Measures

Abstract In this paper, we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the 1st example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups. This paper is dedicated to Alexander Olevskii on the occasion of his birthday. Olevskii’s mathematical depth and personal kindness serve as a major source of inspiration for us and many others in the field of mathematics.

Hadamard triples generate self-affine spectral measures

Let R R be an expanding matrix with integer entries, and let B , L B,L be finite integer digit sets so that ( R , B , L ) (R,B,L) form a Hadamard triple on R d {\mathbb {R}}^d in the sense that the matrix 1 | det R | [ e 2 π i ⟨ R − 1 b , ℓ ⟩ ] ℓ ∈ L , b ∈ B \begin{equation*} \frac {1}{\sqrt {|\det R|}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B} \end{equation*} is unitary. We prove that the associated fractal self-affine measure μ = μ ( R , B ) \mu = \mu (R,B) obtained by an infinite convolution of atomic measures μ ( R , B ) = δ R − 1 B ∗ δ R − 2 B ∗ δ R − 3 B ∗ ⋯ \begin{equation*} \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots \end{equation*} is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in L 2 ( μ ) L^2(\mu ) . This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.

Fourier frames for singular measures and pure type phenomena

Let μ \mu be a positive measure on R d \mathbb {R}^d . It is known that if the space L 2 ( μ ) L^2(\mu ) has a frame of exponentials, then the measure μ \mu must be of “pure type”: it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that μ \mu cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each “pure” component of the measure separately may admit such a frame.

Introducing the windowed Fourier frames technique for obtaining the approximate solution of the coupled system of differential equations

This paper is devoted to introduce an efficient solver using a combination of the symbol of the operator and the windowed Fourier frames (WFFs) of the coupled system of second order ordinary differential equations. The given system has a basic importance in modeling various phenomena like, Cascades and Compartment Analysis, Pond Pollution, Home Heating, Chemostats and Microorganism Culturing, Nutrient Flow in an Aquarium, Biomass Transfer and others. The proposed method reduces the system of differential equations to a system of algebraic equations in the coefficients of WFFs. The introduced method is computer oriented with highly accurate solution. To demonstrate the efficiency of the proposed method, two examples are presented and the results are displayed graphically. Finally, we convert the presented coupled systems of BVPs to a first order system of ODEs to compare the obtained numerical solution with those solutions using the fourth-order Runge–Kutta method (RK4).

Weighted Fourier frames on self-affine measures

Continuing the ideas from our previous paper [6], we construct Parseval frames of weighted exponential functions for self-affine measures.

Translational absolute continuity and Fourier frames on a sum of singular measures

A finite Borel measure μ in Rd is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for L2(μ). It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then ν⁎λ+δt⁎ν is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure μ4+μ16 does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping Rθ with θ≠±π/2, the two-dimensional measure ρθ(⋅):=(μ4×δ0)(⋅)+(δ0×μ16)(Rθ−1⋅), supported on the union of x-axis and y=(cot⁡θ)x, always admit a Fourier frame. Furthermore, we can find {e2πi〈λ,x〉}λ∈Λθ such that it forms a Fourier frame for ρθ with frame bounds independent of θ. Nonetheless, ρ±π/2 does not admit any Fourier frame.

Non-spectral fractal measures with Fourier frames

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.

Detecting Edges from Non-uniform Fourier Data Using Fourier Frames

Edge detection plays an important role in identifying regions of interest in an underlying signal or image. In some applications, such as magnetic resonance imaging (MRI) or synthetic aperture radar (SAR), data are sampled in the Fourier domain. Many algorithms have been developed to efficiently extract edges of images when uniform Fourier data are acquired. However, in cases where the data are sampled non-uniformly, such as in non-Cartesian MRI or SAR, standard inverse Fourier transformation techniques are no longer suitable. Methods exist for handling these types of sampling patterns, but are often ill-equipped for cases where data are highly non-uniform or when the data are corrupted or otherwise not usable in certain parts of the frequency domain. This investigation further develops an existing approach to discontinuity detection, and involves the use of concentration factors. Previous research shows that the concentration factor technique can successfully determine jump discontinuities in non-uniform data. However, as the distribution diverges further away from uniformity so does the efficacy of the identification. Thus we propose a method that employs the finite Fourier approximation to specifically tailor the design of concentration factors. We also adapt the algorithm to incorporate appropriate smoothness assumptions in the piecewise smooth regions of the function. Numerical results indicate that our new design method produces concentration factors which can more precisely identify jump locations than those previously developed in both one and two dimensions.

Weighted Fourier frames on fractal measures

We generalize an idea of Picioroaga and Weber [10] to construct Parseval frames of weighted exponential functions for self-affine measures.

Fourier Frames for the Cantor-4 Set

The measure supported on the Cantor-4 set constructed by Jorgensen–Pedersen is known to have a Fourier basis, i.e. that it possess a sequence of exponentials which form an orthonormal basis. We construct Fourier frames for this measure via a dilation theory type construction. We expand the Cantor-4 set to a two dimensional fractal which admits a representation of a Cuntz algebra. Using the action of this algebra, an orthonormal set is generated on the larger fractal, which is then projected onto the Cantor-4 set to produce a Fourier frame.

Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes.

Nonharmonic nonlinear Fourier frames and convergence of corresponding frame series

Having redundancy, frames, compared with basis, can provide more robust representation of a vector in application. By introducing nonharmonic nonlinear Fourier frames, a method is established to construct such frames by perturbation. Based on a special class of nonharmonic nonlinear Fourier frames, the convergence of its corresponding frame operator is investigated, and the convergence rate, associated with the coefficient sequence of the frame operator, is estimated. Finally, we also discuss the equiconvergence of two different (nonlinear or linear) Fourier (basis or frame) series of f $\in$ L2 (−π,π).

Windowed Fourier Frames to Approximate Two-Point Boundary Value Problems

Boundary value problems arise while modeling various physical and engineering reality. In this communication we investigate windowed Fourier frames focusing two-point BVPs. We approximate BVPs using windowed Fourier frames. We present some numerical results to demonstrate the efficiency of such frame functions to approximate BVPs.

Generalized Fourier Frames in Terms of Balayage

We develop a theory of non-uniform sampling in the context of the theory of frames for the settings of the short time fourier transform and pseudo-differential operators. Our theory is based on profound historical precedents including Beurling’s theory of balayage, emanating from the nineteenth century work of Christoffel and Poincare, the theory and results from spectral synthesis due to Wiener and Beurling and a host of the major harmonic analysts of the twentieth century, and the theory of sets of multiplicity, going back to Riemann and emerging fundamentally from the Russian school of harmonic analysis in the early twentieth century. Our results are meant to serve as the underpinnings for both theoretical and practical results in the realm of non-uniform sampling. They can also be compared with several other distinct forays into non-uniform sampling, including the settings of quasi-crystals and modulation spaces, where proofs for the latter setting require the analysis of convolution operators on the Heisenberg group. Our theory herein is the first step in which the ultimate goal is computational implementation for non-uniform sampling and its myriad applications, where balayage, spectral synthesis, and sets of multiplicity are computationally quantified. A critical component is to resurrect the formulation of balayage in terms of covering criteria.

Uniformity of measures with Fourier frames

We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. To be more precise, by considering certain absolute continuity properties of the measure and its translation, we recover the characterization on absolutely continuous measures gdx with Fourier frames obtained in [24]. Moreover, we prove that the frame bounds are pushed away by the essential infimum and supremum of the function g. This also shows that absolutely continuous spectral measures supported on a set Ω, if they exist, must be the standard Lebesgue measure on Ω up to a multiplicative constant. We then investigate affine iterated function systems (IFSs), we show that if an IFS with no overlap admits a frame measure then the probability weights are all equal. Moreover, we also show that the Łaba–Wang conjecture [20] is true if the self-similar measure is absolutely continuous. Finally, we will present a new approach to the conjecture of Liu and Wang [29] about the structure of non-uniform Gabor orthonormal bases of the form G(g,Λ,J).

Frames of multi-windowed exponentials on subsets of [formula omitted

Given discrete subsets Λj⊂Rd, j=1,…,q, consider the set of windowed exponentials ⋃j=1q{gj(x)e2πi〈λ,x〉:λ∈Λj} on L2(Ω). We show that a necessary and sufficient condition for the windows gj to form a frame of windowed exponentials for L2(Ω) with some Λj is that 0<m⩽maxj∈J|gj|⩽M almost everywhere on Ω for some subset J of {1,…,q}. If Ω is unbounded, we show that there is no frame of windowed exponentials if the Lebesgue measure of Ω is infinite. If Ω is unbounded but of finite measure, we give a simple sufficient condition for the existence of Fourier frames on L2(Ω). At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.

Continuous and discrete Fourier frames for fractal measures

Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on R 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 measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.

On Characterizations of Fourier Frames and Tilings

We give some characterizations of Fourier frames and tilings and obtain a more general form of characterizations of spectra and tilings.

Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations

We investigate the application of windowed Fourier frames to the numerical solution of partial differential equations, focussing on elliptic equations. The action of a partial differential operator (PDO) on a windowed plane wave is close to a multiplication, where the multiplication factor is given by the symbol of the PDO evaluated at the wave number and central position of the windowed plane wave. This can be exploited in a preconditioning method for use in iterative inversion. For domains with periodic boundary conditions we find that the condition number with the preconditioning becomes bounded and the iteration converges well. For problems with a Dirichlet boundary condition, some large and small singular values remain. However the iterative inversion still appears to converge well.