Fourier frames on smooth surfaces with nonvanishing Gaussian curvature

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

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.

Interleaving spirals are a natural setting for attaining fast MRI signal reconstruction. Using the results of Beurling and Landau, as well as quantitative coverings of the spectral domain by translations of the polar set of the target disk space E, harmonics for Fourier frames F are constructed on interleaving spirals to reconstruct signals on E. Because of the weak Fourier frame bound estimates by standard calculations, implementation is addressed in terms of finite frame approximants of F.

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.

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

On the Beurling dimension of exponential frames

Motivated by recent results on non-vanishing spatial curvature [1] we employ aholographic model of dark energy to investigate the validity of the first and second laws ofthermodynamics in a non-flat (closed) universe enclosed by the apparent horizonRA and the event horizon measured from the sphere of the horizon namedL. We show that for the apparent horizon the first law is roughly respected fordifferent epochs while the second law of thermodynamics is respected, while forL as the system’s IR cut-off the first law is broken and the second law is respected for thespecial range of the deceleration parameter. It is also shown that for the late-time universeL is equalto RA and the thermodynamic laws hold when the universe has non-vanishing curvature. Definingthe fluid temperature as being proportional to the horizon temperature the range for thecoefficient of proportionality is obtained, provided that the generalized second law ofthermodynamics holds.

We solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of L^p-spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

Gabor orthogonal bases and convexity, Discrete Analysis 2018:19, 11 pp. A fundamental way of understanding a function $f$ defined on $\mathbb R^d$ is to expand it in terms of a basis with nice properties. Typically, one assumes that $f\in L_2(\mathbb R^d)$, and then it becomes natural to look for orthonormal bases with properties such as interesting symmetries. For example, wavelet bases, which play a very important role in signal processing, are orthonormal and consist of translates and dilates of a single function. One class of good bases that has been studied is the class of _Gabor orthogonal bases_. Here the idea is to take a countable set of functions of the form $g_{a,b}(x)=g(x-a)\exp(-2\pi ix.b)$, where $g$ is a fixed function in $L_2(\mathbb R^d)$ and $a,b$ are elements of $\mathbb R^d$. One way to achieve this is to find a measurable set $K$ that tiles $\mathbb R^d$ with the property that there is an orthonormal basis of $L_2(K)$ that consists of functions of the form $\chi_K(x)\exp(-2\pi ix.b)$. Then we can obtain a similar orthonormal basis for $L_2(K+t)$ for each translate $t$ in the tiling, and combining these bases one obtains a basis for the whole of $L_2(\mathbb R^d)$, for the simple reason that there is no interaction between the bases for the different translates. It was conjectured by Fuglede in 1974 that a set $K$ admits an orthonormal basis of exponentials as above if and only if it tiles $\mathbb R^d$, but this was shown not to be true by Tao in 2003. Understanding which sets are spectral in this way is still an active area of research. In general, our understanding of which functions $g$ can be used to form Gabor orthogonal bases is quite limited. In particular, it seems to be hard to prove negative results. In this paper, it is shown that a certain class of functions cannot be used: if $K$ is a convex body with a smooth boundary with Gaussian curvature that does not vanish anywhere, then provided that $d\ne 1 \mod 4$, the characteristic function of $K$ cannot be used as the function $g$ in a Gabor orthogonal basis. Of course, there is no chance of tiling $\mathbb R^d$ with such a set $K$, but that on its own does not rule out some kind of complicated interaction between neighbouring local bases. The condition that the Gaussian curvature does not vanish everywhere is a strong one, but the authors believe that it should not be too hard to modify the proof to do without it, since there will be plenty of points with non-vanishing curvature even if not all of them have it. The condition that $d\ne 1\mod 4$ is more mysterious -- late on in the proof, a contradiction is obtained, which fails to be a contradiction if $d=1\mod 4$, but that appears to be an artefact of the proof rather than a serious indication that the result is different in that case. More or less all the obvious questions one might want to ask about Gabor orthogonal bases are still open. For example, it is not known whether there is a non-spectral set $K$ such that $\chi_K$ will work. In this case, one would not be able to find an orthonormal basis of exponentials for $L_2(K)$, so one would have to combine non-spanning orthonormal bases of exponentials for overlapping translates of $K$. A particularly interesting case that the authors mention is that of a triangle in $\mathbb R^2$.

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ⊂ ℤ+ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials.Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [−q, q]2) ≦ C(K) q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L 2(K) does not possess an orthogonal basis of exponentials.

A roughness measurement method based on the application of a photodiode integrator for scattered radiation gathering as well as results of experiments concerning this method are presented in the paper. The new integrator, contrary to known optical integrating devices, provides the possibility to get much less expensive and smaller instruments than traditional ones. Unfortunately, a decrease of the integrator dimensions could restrict its spatial frequency bandwidth causing measurement errors. Therefore, experimental works have been performed to find the relation between the range-of-acceptance angle of the integrator and measured rms roughness. They have been done by constructing a model of the integrator and a complete TIS instrument as well as a precision experimental set. For the purpose of making a direct comparison between the new instrument and the existing one, the Ulbricht sphere has also been used in the set. For very smooth surfaces, the value of the lower angle range is very critical and should be smaller than 2 deg. The upper limit can be about 30 deg. for measurement of smooth and isotropic surfaces. At such limits, the influence of the integrator flatness is stated to be non-substantial. Results of surface roughness measurements obtained from the new unit and the Ulbricht sphere show that both methods have similar issues. Taking into account the dimensions of the photodiode integrator, a small and inexpensive TIS instrument can be designed for measuring roughness and reflectance of isotropic smooth surfaces in the optical and electronic industries.

The sliding behavior of droplets on smooth and rough surfaces with various surface wettabilities is investigated by many-body dissipative particle dynamics simulations. On a smooth surface, as the driving force (Bo) increases, the droplet shape and velocity (Cac) before breakage can be classified into four distinct regimes: (I) nearly spherical cap with Cac∝Bo; (II) oval shape with negative deviation from the linear relation; (III) elongated shape without a neck, where Cac decreases with increasing Bo; and (IV) oscillation of an elongated shape with fluctuating sliding velocity. On rough surfaces, corner-shaped droplets, which are absent on a smooth surface, can be observed. A further increase in Bo leads to the formation of cusp and pearling. Different from pinching-off on rough surfaces, which produces a cascade of smaller droplets through groove-induced shedding, chaotic breakage of a droplet on a smooth surface is caused by an unsteady flow field. Finally, a universal linear relationship between the sliding velocity based on the surface velocity (Cas) and the modified driving force (Bo**) is derived to take into account the effects of surface wettability and roughness.

This research aims to 1) Create a commutator smooth surface measurement by using the laser displacement sensor. 2) Automatically correct and store data. 3) Develop a computer program for measuring the surface of commutator. The research methods are divided into two stages. First, design a commutator smooth surface measurement by using the laser displacement sensor and built base station in order to adjust position of the laser displacement sensor. Additionally, we create a set of motor-drive automation to stabilize the commutator speed by installing the encoder Incremental to obtain the position and angle. Second, we develop a computer program for measuring the commutator surface in real-time and compare our results of the smooth commutator surface in a polar form with commutator surface reference graphs. The results showed that, our commutator surface measuring set effectively measured the smooth surface and fault models. Moreover, our design program effectively displayed in linear graphs and polar graphs the damaged or worn surface of the commutator.

This study demonstrates a comparative analysis of surface and satellite measurements. The average concentrations of PM_2.5 and CO, SO₂, measured at the Listvyanka station located on the coast of Lake Baikal, were considered. Satellite measurements data (Copernicus Sentinel-5P) were recomputed based on the SILAM model. A joint analysis of data showed that satellite measurements were suitable for a spatial description of regional air pollution. The computed maxima coincided with the surface measurements in terms of time periods and general monitoring results. However, at extreme increases in concentrations of pollutants, a significant difference in the numerical values was registered. Satellite monitoring data confirmed the relationship between the increase in PM_2.5 and CO concentrations in the air basin at the Listvyanka station and the transfer of smoke plumes from intense forest fires located at a distance of 1,500 – 2,000 km.