Fourier Frame Research Articles

A least-squares Fourier frame method for nonlocal diffusion models on arbitrary domains

We introduce a least-squares Fourier frame method for solving nonlocal diffusion models with Dirichlet volume constraint on arbitrary domains. The mathematical structure of a frame rather than a basis allows using a discrete least-squares approximation on irregular domains and imposing non-periodic boundary conditions. The method has inherited the one-dimensional integral expression of Fourier symbols of the nonlocal diffusion operator from Fourier spectral methods for any d spatial dimensions. High precision of its solution can be achieved via a direct solver such as pivoted QR decomposition even though the corresponding system is extremely ill-conditioned, due to the redundancy in the frame. The extension of AZ algorithm improves the complexity of solving the rectangular linear system to O(Nlog2⁡N) for 1d problems and O(N2log2⁡N) for 2d problems, compared with O(N3) of the direct solvers, where N is the number of degrees of freedom. We present ample numerical experiments to show the flexibility, fast convergence and asymptotical compatibility of the proposed method.

(p, q)-Frame measures on LCA groups: Perturbations and construction

Motivated to generalize the Fourier frame concept to Banach spaces we introduce (p, q)-Bessel/frame measures for a given finite measure on LCA groups. We also present a general way of constructing (p, q)-Bessel/frame measures for a given measure. Moreover, we prove that if a measure has an associated (p, q)-frame measure, then it must have a certain uniformity in the sense that the weight is distributed quite uniformly on its support. Next, we show that if the measures μ and λ without atoms whose supports form a packing pair, then μ⁎λ+δg⁎μ does not admit any (p, q)-frame measure. Finally, we analyze the stability of (p, q)-frame measures under small perturbations. We prove new theorems concerning the stability of (p, q)-frame measures under perturbation in both Hilbert spaces and Banach spaces.

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.

Positive matrices in the Hardy space with prescribed boundary representations via the Kaczmarz algorithm

For a singular probability measure μ on the circle, we show the existence of positive matrices on the unit disc which admit a boundary representation on the unit circle with respect to μ. These positive matrices are constructed in several different ways using the Kaczmarz algorithm. Some of these positive matrices correspond to the projection of the Szegő kernel on the disc to certain subspaces of the Hardy space corresponding to the normalized Cauchy transform of μ. Other positive matrices are obtained which correspond to subspaces of the Hardy space after a renormalization, and so are not projections of the Szegő kernel. We show that these positive matrices are a generalization of a spectrum or Fourier frame for μ, and the existence of such a positive matrix does not require μ to be spectral.

Early Embryo Development in Mares: Proteomics of Uterine Fluid

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.

Finite Fourier Frame Approximation Using the Inverse Polynomial Reconstruction Method

In several applications, data are collected in the frequency (Fourier) domain non-uniformly, either by design or as a consequence of inexact measurements. The two major bottlenecks for image reconstruction from non-uniform Fourier data are (i) there is no obvious way to perform the numerical approximation, as the non-uniform Fourier data is not amenable to fast transform techniques and resampling the data first to uniform spacing is often neither accurate or robust; and (ii) the Gibbs phenomenon is apparent when the underlying function (image) is piecewise smooth, an occurrence in nearly every application. Recent investigations suggest that it may be useful to view the non-uniform Fourier samples as Fourier frame coefficients when designing reconstruction algorithms that attempt to mitigate either of these fundamental problems. The inverse polynomial reconstruction method (IPRM) was developed to resolve the Gibbs phenomenon in the reconstruction of piecewise analytic functions from spectral data, notably Fourier data. This paper demonstrates that the IPRM is also suitable for approximating the finite inverse Fourier frame operator as a projection onto the weighted $$L_2$$ space of orthogonal polynomials. Moreover, the IPRM can also be used to remove the Gibbs phenomenon from the Fourier frame approximation when the underlying function is piecewise smooth. The one-dimensional numerical results presented here demonstrate that using the IPRM in this way yields a robust, stable, and accurate approximation from non-uniform Fourier data.

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.

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.

Tight frames, partial isometries, and signal reconstruction

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for the space of square integrable functions whose domain is the ball of radius When a finite number of measurements is used to reconstruct a signal in error estimates arising from such approximation are discussed.

A Frame Theoretic Approach to the Nonuniform Fast Fourier Transform

Nonuniform Fourier data are routinely collected in applications such as magnetic resonance imaging, synthetic aperture radar, and synthetic imaging in radio astronomy. To acquire a fast reconstruction that does not require an online inverse process, the nonuniform fast Fourier transform (NFFT), also called convolutional gridding, is frequently employed. While various investigations have led to improvements in accuracy, efficiency, and robustness of the NFFT, not much attention has been paid to the fundamental analysis of the scheme, and in particular its convergence properties. This paper analyzes the convergence of the NFFT by casting it as a Fourier frame approximation. In so doing, we are able to design parameters for the method that satisfy conditions for numerical convergence. Our so-called frame theoretic convolutional gridding algorithm can also be applied to detect features (such as edges) from nonuniform Fourier samples of piecewise smooth functions.

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 Fourier frame of absolutely continuous measures

Let μ be a compactly supported absolutely continuous probability measure on R n , we show that L 2 ( K , d μ ) admits a Fourier frame if and only if its Radon–Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if μ is an equal weight absolutely continuous self-similar measure on R 1 and L 2 ( K , d μ ) admits a Fourier frame, then the density of μ must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1 / 2 < λ < 1 , the L 2 space of the λ-Bernoulli convolutions cannot admit a Fourier frame.

Syndetic sets, paving and the Feichtinger conjecture

We prove that if a Bessel sequence in a Hilbert space that is indexed by a countably infinite group in an invariant manner can be partitioned into finitely many Riesz basic sequences, then each of the sets in the partition can be chosen to be syndetic. We then apply this result to prove that if a Fourier frame for a measurable subset of a higher dimensional cube can be partitioned into Riesz basic sequences, then each subset can be chosen to be a syndetic subset of the corresponding higher dimensional integer lattice. Both of these results follow from a result about syndetic pavings of elements of the von Neumann algebra of a discrete group.

On the Beurling dimension of exponential frames

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff dimension of the fractal.

A Comparative Evaluation of Techniques for Single-Frame Discrimination of Nonstationary Sinusoids

Many spectral analysis and modification techniques require the separation of sinusoidal from nonsinusoidal signal components of a Fourier spectrum. Techniques exist for the estimation of the parameters of nonstationary sinusoids, and for discriminating these from other components, within a single Fourier frame. We present a comparative study of five methods for sinusoidal discrimination, considering their effectiveness and their computational cost.

Illumination Compensation for Wave Equation Migration Shadow

AbstractDue to the effects of complicated subsurface structures and seismic acquisition system, the seismic wave illumination to some subsurface targets may be inhomogeneous, so the seismic acquisition system can not effectively acquire the reflected information from these targets, and migrated images on these targets will have some migration shadows. Based on the windowed Fourier frame expansion of wavefield and Green function, an illumination compensation method for wave equation migration shadow is proposed by using the wave equation migration and illumination analysis in angle domain and combining the wave equation inverse theory. The compensation method can take into account the effects of the seismic acquisition system and wave propagation path, remove the migration shadow on the complicated subsurface structures, and improve the results of wave equation prestack depth migration in complex structure regions.

Delay-coordinates embeddings as a data mining tool for denoising speech signals

In this paper, we utilize techniques from the theory of nonlinear dynamical systems to define a notion of embedding estimators. More specifically, we use delay-coordinates embeddings of sets of coefficients of the measured signal (in some chosen frame) as a data mining tool to separate structures that are likely to be generated by signals belonging to some predetermined data set. We implement the embedding estimator in a windowed Fourier frame, and we apply it to speech signals heavily corrupted by white noise. Our experimental work suggests that, after training on the data sets of interest, these estimators perform well for a variety of white noise processes and noise intensity levels.

The analysis and design of windowed Fourier frame based multiple description source coding schemes

In this paper the windowed Fourier encoding-decoding scheme applied to the multiple description compression problem is analyzed. In the general case, four window functions are needed to define the encoder and decoder, although this number can be reduced to three or two by using time-shift or frequency-shift division schemes. The encoding coefficients are next divided into two groups according to the eveness of either the modulation or translation index. The distortion on each channel is analyzed using the Zak transform. For the optimal windows, explicit representation formulas are obtained and nonlocalization results are proved. Asymptotic formulas of the total distortion and transmission rate are established and the redundancy is shown to trade off between these two.

Applications of time-frequency processing to radar imaging

Due to the time-varying behavior of the Doppler frequency of radar returns, and due to the multiple backscattering behavior of radar targets, the resolution of radar images can be significantly degraded and those images may be blurred. Conventional radar processors use the Fourier transform to retrieve Doppler information. To use the Fourier transform properly, some restrictions must be applied: the scatterers must remain in their range cells and their Doppler frequency shifts should be stationary during the entire imaging time. However, due to a target’s complex motion, the Doppler frequency shifts will be timevarying. Therefore, the Doppler spectrum obtained from the Fourier transform will be smeared, and, the radar image will be blurred. However, the restrictions of the Fourier transform can be lifted if the Doppler information is retrieved with a time-frequency transform that does not require a stationary Doppler spectrum. The image blurring problem caused by time-varying Doppler frequency shifts can be solved without resorting to sophisticated motion-compensation techniques. By replacing the conventional Fourier transform with a time-frequency transform, a 2-D range-Doppler Fourier frame becomes a 3-D time-range-Doppler cube. By sampling in time, a time sequence of 2-D range-Doppler images can be viewed. Individual, time-sampled images from the cube provide superior image resolution. When targets contain cavities or ducttype structures, these structures’ scattering mechanisms appear in radar images as blurred ‘‘clouds’’ extending in range. It is beneficial to incorporate the time-frequency transform into range profiles of the radar image. By so doing ‘‘clouds’’ can be removed and structure resonance frequencies identified.