Recent zbMATH articles in MSC 42Ahttps://zbmath.org/atom/cc/42A2021-06-15T18:09:00+00:00WerkzeugColor image restoration and inpainting via multi-channel total curvature.https://zbmath.org/1460.940142021-06-15T18:09:00+00:00"Tan, Lu"https://zbmath.org/authors/?q=ai:tan.lu"Liu, Wanquan"https://zbmath.org/authors/?q=ai:liu.wanquan"Pan, Zhenkuan"https://zbmath.org/authors/?q=ai:pan.zhenkuanSummary: The multi-channel total variation (MTV) based on L2 norm is capable of preserving object edges and smoothing flat regions in color images. However, it will lead to loss of image contrast, smear object corners, and produce staircase artifacts in the restored images. In order to remedy these side effects, we propose a new multi-channel total curvature model based on L1 norm (MTC-L1) for vector-valued image restoration in this paper. By introducing some auxiliary variables and Lagrange multipliers, we develop a fast algorithm based alternating direction method of multipliers (ADMM) for the proposed model, which allows the use of the fast Fourier transform (FFT), generalized soft threshold formulas and projection method. Extensive experiments have been conducted on both synthetic and real color images, which validate the proposed approach for better restoration performance, and show advantages of the proposed ADMM over algorithms based on traditional gradient descent method (GDM) in terms of computational efficiency.Remarks on the uniform and absolute convergence of orthogonal series.https://zbmath.org/1460.420432021-06-15T18:09:00+00:00"Kashin, B. S."https://zbmath.org/authors/?q=ai:kashin.boris-sFor finite orthonormal systems it is investigated, whether it is possible to improve the properties of a function by changing a few of its Fourier coefficients. The other question studied in this paper is the construction of examples of smooth functions the multi-dimensional Fourier series of which are not absolutely convergent.
Reviewer: Ferenc Weisz (Budapest)Irregularity of distribution in Wasserstein distance.https://zbmath.org/1460.111052021-06-15T18:09:00+00:00"Graham, Cole"https://zbmath.org/authors/?q=ai:graham.coleThe author considers the non-uniformity of probability measures
on the (unit) interval and the circle, by making use of the so-called Wasserstein-\(p\) distance.
To be more precise, let \(X\) be equal to \([0,1)\) or to \(\mathbb{R}/\mathbb{Z}\), and let \(\lambda\)
be the uniform measure on \(X\). Furthermore, let \(\mathcal{P} (X)\) be the space of probability measures
on \(X\). For \(p\in [1,\infty]\) and \(\mu\in\mathcal{P} (X)\), the Wasserstein-\(p\) distance between \(\lambda\) and
\(\mu\) is defined as
\[
W_p^X (\mu,\lambda):=\| \mu - \lambda \|_{\dot{W}^{-1,p}(X)},
\]
where \(\|\cdot\|_{\dot{W}^{-1,p}(X)}\) is a negative Sobolev norm.
Now, for a given sequence of points \((x_n)_{n\in\mathbb{N}}\) in \([0,1)\) and \(N\in \mathbb{N}\), define
\[
\mu_N:=\frac{1}{N}\sum_{n=1}^N \delta_{x_n}.
\]
One of the main results in the paper is the following estimate. For \(X\) as above, for every \(p\in [1,\infty]\) there
exists a positive constant \(C_p\) such that for any sequence of points \((x_n)_{n\in \mathbb{N}}\) in \(X\) it is true that
\[
W_p^X (\mu_N,\lambda)\ge C_p \frac{(\log N)^{\alpha_p}}{N}
\]
holds for infinitely many \(N\in\mathbb{N}\). Furthermore, this bound is sharp. Here, \(\alpha_p\) is \(1/2\) for finite \(p\),
and 1 if \(p=\infty\).
Moreover, the author shows an explicit upper bound on non-uniformity, and applies these findings to the equidistribution
of quadratic residues in finite fields.
Reviewer: Peter Kritzer (Linz)The sharp time decay rate of the isentropic Navier-Stokes system in \(\mathbb{R}\).https://zbmath.org/1460.352522021-06-15T18:09:00+00:00"Chen, Yuhui"https://zbmath.org/authors/?q=ai:chen.yuhui"Pan, Ronghua"https://zbmath.org/authors/?q=ai:pan.ronghua"Tong, Leilei"https://zbmath.org/authors/?q=ai:tong.leileiSummary: We investigate the sharp time decay rates of the solution \(U\) for the compressible Navier-Stokes system (1.1) in \(\mathbb{R}^3\) to the constant equilibrium \((\bar\rho>0,0)\) when the initial data is a small smooth perturbation of \((\bar\rho, 0)\). Let \(\widetilde{U}\) be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that \(\|U-\widetilde{U}\|_{L^2}\) decays at least at the rate of \((1+t)^{-\frac{5}{4}}\), which is faster than the rate \((1+t)^{-\frac{3}{4}}\) for the \(\widetilde{U}\) to its equilibrium \((\bar\rho,0)\). Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.Energy bands and Wannier functions of the fractional Kronig-Penney model.https://zbmath.org/1460.810212021-06-15T18:09:00+00:00"Vellasco-Gomes, Arianne"https://zbmath.org/authors/?q=ai:gomes.arianne-vellasco"de Figueiredo Camargo, Rubens"https://zbmath.org/authors/?q=ai:de-figueiredo-camargo.rubens"Bruno-Alfonso, Alexys"https://zbmath.org/authors/?q=ai:bruno-alfonso.alexysSummary: Energy bands and Wannier functions of the fractional Schrödinger equation with a periodic potential are calculated. The kinetic energy contains a Riesz derivative of order \(\alpha\), with \(1 <\alpha \leq 2\), and numerical results are obtained for the Kronig-Penney model. Bloch and Wannier functions show cusps in real space that become sharper as \(\alpha\) decreases. Energy bands and Bloch functions are smooth in reciprocal space, except at the \(\Gamma\) point. Depending on symmetry, each Wannier function decays as a power-law with exponent \(-(\alpha +1)\) or \(-(\alpha +2)\). Closed forms of their asymptotic behaviors are given. Each higher band displays anomalous behavior as a function of potential strength. It first narrows, becoming almost flat, then widens, with its width tending to a constant. The position uncertainty of each Wannier function follows a similar trend.Approximations on classes of Poisson integrals by Fourier-Chebyshev rational integral operators.https://zbmath.org/1460.420032021-06-15T18:09:00+00:00"Potseiko, P. G."https://zbmath.org/authors/?q=ai:potseiko.p-g"Rovba, E. A."https://zbmath.org/authors/?q=ai:rovba.e-aSummary: Introducing some classes of the functions defined by Poisson integrals on the segment \([-1,1]\) and studying approximations by Fourier-Chebyshev rational integral operators on the classes, we establish integral expressions for approximations and upper bounds for uniform approximations. In the case of boundary functions with a power singularity on \([-1,1] \), we find the upper bounds for pointwise and uniform approximations and an asymptotic expression for a majorant of uniform approximations in terms of rational functions with a fixed number of prescribed geometrically distinct poles. Considering two geometrically distinct poles of the approximant of even multiplicity, we obtain asymptotic estimates for the best uniform approximation by this method with a higher convergence rate than polynomial analogs.Introduction to the discrete Fourier series considering both mathematical and engineering aspects. A linear-algebra approach.https://zbmath.org/1460.940242021-06-15T18:09:00+00:00"Kohaupt, Ludwig"https://zbmath.org/authors/?q=ai:kohaupt.ludwigSummary: The discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating images in computer tomography. In order to achieve this, appropriate algorithms are necessary. In this context, the fast Fourier transform (FFT) plays a key role which is an algorithm for calculating the discrete Fourier transform (DFT); this, in turn, is tightly connected with the discrete Fourier series. The last one itself is the discrete analog of the common (continuous-time) Fourier series and is usually learned by mathematics students from a theoretical point of view. The aim of this expository/pedagogical paper is to give an introduction to the discrete Fourier series for both mathematics and engineering students. It is intended to expand the purely mathematical view; the engineering aspect is taken into account by applying the FFT to an example from signal processing that is small enough to be used in class-room teaching and elementary enough to be understood also by mathematics students. The MATLAB program is employed to do the computations.New characterizations for the multi-output correlation-immune Boolean functions.https://zbmath.org/1460.941002021-06-15T18:09:00+00:00"Chai, Jinjin"https://zbmath.org/authors/?q=ai:chai.jinjin"Mesnager, Sihem"https://zbmath.org/authors/?q=ai:mesnager.sihem"Wang, Zilong"https://zbmath.org/authors/?q=ai:wang.zilongSummary: Correlation-immune (CI) multi-output Boolean functions have the property of keeping the same output distributions when some input variables are fixed. Recently, a new application of CI functions has appeared in the system of resisting side-channel attacks (SCA). In this paper, three new methods are proposed to characterize the \(t\)-th order CI multi-output (\(n\)-input and \(m\)-output) Boolean functions. The first characterization is to regard the multi-output Boolean functions as the corresponding generalized Boolean functions. It is shown that a generalized Boolean function \(f_g\) is a \(t\)-th order CI function if and only if the Walsh transform of \(f_g\) defined here vanishes at all points with Hamming weights between 1 and \(t\). The last two methods are generalized from Fourier spectral characterizations. Especially, Fourier spectral characterizations are efficient to characterize the symmetric multi-output CI Boolean functions.Time-frequency transform involving nonlinear modulation and frequency-varying dilation.https://zbmath.org/1460.420072021-06-15T18:09:00+00:00"Chen, Qiuhui"https://zbmath.org/authors/?q=ai:chen.qiuhui"Li, Luoqing"https://zbmath.org/authors/?q=ai:li.luoqing"Qian, Tao"https://zbmath.org/authors/?q=ai:qian.taoSummary: This paper designs a general type time-frequency transform whose kernel function involves a nonlinear modulation and a frequency-varying dilation. The corresponding inversion formula is established.The Nyquist sampling rate for spiraling curves.https://zbmath.org/1460.420492021-06-15T18:09:00+00:00"Jaming, Philippe"https://zbmath.org/authors/?q=ai:jaming.philippe"Negreira, Felipe"https://zbmath.org/authors/?q=ai:negreira.felipe"Romero, José Luis"https://zbmath.org/authors/?q=ai:romero.jose-luisSummary: We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis.Orthogonal polynomials, biorthogonal polynomials and spline functions.https://zbmath.org/1460.410042021-06-15T18:09:00+00:00"Goh, Say Song"https://zbmath.org/authors/?q=ai:goh.say-song"Goodman, Tim N. T."https://zbmath.org/authors/?q=ai:goodman.timothy-n-t"Lee, S. L."https://zbmath.org/authors/?q=ai:lee.seunggyu-lee|lee.seok-lyong|lee.ssu-lang|lee.sung-l|lee.shao-lin|lee.seng-luan|lee.su-ling|lee.shen-ling|lee.shyi-long|lee.seng-lip|lee.shong-leih|lee.steven-lFor orthogonal and bi-orthogonal polynomials, there are several important concepts that the authors of this article succeed to generalise. Among them are so-called generating functions and the Rodrigues' formula. Concretely, the authors wish to replace the expressions \(f_m\) in the Rodrigues' formula which amount to weight-functions \(\omega\) times some simple powers (often, monomials or even constants, or at most quadratics anyway, that are then taken to \(m\)-th powers) in the classical cases, by B-splines. The \(f_m\) are assumed to be decaying exponentially, as \(O(\exp(-Bx-\epsilon x))\). As a result, their Fourier transforms become analytic in an open strip of size \(B\).
For comparison, for Laguerre polynomials for example, the \(f_m\) would be multiples of an exponential weight function times \(x^m\), for Hermite weight function \(\exp(-x^2)\) (times \(1^m\)), and these are now replaced by B-splines.
Both uniform B-splines (pen-ultimate section) and non-uniform knots (last section of the article) are treated, and there is even a further generalisation where the B-splines are defined by repeated convolution not starting from a characteristic function of an interval (the classical case), but a less restricted function as an initial value. The Fourier transforms of those ``generalised B-Splines'' alter in that the usual powers of sinc-functions get another factor of the Fourier transform of the starting (initial) function.
The result is called a generalised Rodrigues' formula and the needed generating function is called a generalised generating function. In order to form the generalised generating function, Fourier transforms for the \(f_m\) are needed which would be simply convolutions of weight functions with some derivatives of \(\delta\)-functions in the classical case (i.e., derivatives of the weight function). This reformulated Rodrigues' formula \(\mu_m=(-1)^m f_m^{(m)}\) is equivalent to considering the \(\mu_m\)s as bi-orthogonal to a sequence of orthogonal polynomials \(Q_m\) that are one factor of the expansion coefficients of the generalised generating function; the other factor being the Fourier transforms of the \(f_m\): \[\exp(xz)=\sum\nolimits_0^\infty Q_m(x) z^m \hat f_m(\mathrm{i}z).\] The main results guarantee the existence of the \(Q_m\) polynomials and the well-definedness of the generalised generating function (these results are, in particular, Theorem~2.3 and Theorem~3.1 and -- for the most general case -- Theorem~4.3).
Reviewer: Martin D. Buhmann (Gießen)On optimal autocorrelation inequalities on the real line.https://zbmath.org/1460.420022021-06-15T18:09:00+00:00"Madrid, José"https://zbmath.org/authors/?q=ai:madrid.jose-a-jimenez"Ramos, João P. G."https://zbmath.org/authors/?q=ai:ramos.joao-p-gSummary: We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work [\textit{R. C. Barnard} and \textit{S. Steinerberger}, J. Number Theory 207, 42--55 (2020; Zbl 1447.11008)]. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove existence of extremizers to these inequalities in certain specific settings. Our methods consist of relating the inequalities in question to other classical sharp inequalities in Fourier analysis, such as the sharp Hausdorff-Young inequality, and employing functional analysis as well as measure theory tools in connection to a suitable dual version of the problem to identify and impose conditions on extremizers.Spectrality of self-affine Sierpinski-type measures on \(\mathbb{R}^2\).https://zbmath.org/1460.420082021-06-15T18:09:00+00:00"Dai, Xin-Rong"https://zbmath.org/authors/?q=ai:dai.xinrong"Fu, Xiao-Ye"https://zbmath.org/authors/?q=ai:fu.xiaoye"Yan, Zhi-Hui"https://zbmath.org/authors/?q=ai:yan.zhi-huiSummary: In this paper, we study the spectral property of a class of self-affine measures \(\mu_{R,\mathcal{D}}\) on \(\mathbb{R}^2\) generated by the iterated function system \(\{\phi_d(\cdot)=R^{-1}(\cdot+d)\}_{d\in\mathcal{D}}\) associated with the real expanding matrix \(R=\begin{pmatrix} b_1 & 0 \\ 0 & b_2\end{pmatrix}\) and the digit set \(\mathcal{D}=\{\binom{0}{0},\binom{1}{0},\binom{0}{1}\}\). We show that \(\mu_{R,\mathcal{D}}\) is a spectral measure if and only if \(3|b_i\), \(i=1,2\). This extends the result of \textit{Q.-R. Deng} and \textit{K.-S. Lau} [J. Funct. Anal. 269, No. 5, 1310--1326 (2015; Zbl 1323.28011)], where they considered the case \(b_1=b_2\). And we also give a tree structure for any spectrum of \(\mu_{R,\mathcal{D}}\) by providing a decomposition property on it.A short glimpse of the giant footprint of Fourier analysis and recent multilinear advances.https://zbmath.org/1460.420012021-06-15T18:09:00+00:00"Grafakos, Loukas"https://zbmath.org/authors/?q=ai:grafakos.loukasIndeed, the paper provides a concise but to-the-point overview of the genesis and influence of Fourier series and Fourier analysis, not only in itself in mathematics but also in physics, like finding \textit{temperatures}, \textit{radio telecommunications}, \textit{acoustics}, \textit{oceanography},\textit{optics}, \textit{spectroscopy}, \textit{crystallography}. [The words in Italics are the fields of investigations mentioned by the author.] Also, he says that Fourier series nowadays are very important for \textit{signal} and \textit{image processing}. Developments are, in short, mentioned for \textit{wavelets}. As to mathematics, he mentions: summation of Fourier series in higher dimensions, products of Fourier series. To close with, the author indicates some advances in multilinear Fourier analysis.
In the bibliography, one finds for instance, papers of the author himself, Carleson, Fefferman, Leong, Farkas, Kahane, Konyagin, Lacey, Thiele, Riesz etc. As to the fundamental work about wavelets due to Ingrid Daubechies [not at all mentioned by the author] one can best consult [\textit{D. Huylebrouck}, ``Ingrid Daubechies, JPEG inventrix and first female professor in Princeton'', in: België + Wiskunde. Gent: Academia Press. 35--70 (2013)].
For the entire collection see [Zbl 1433.00042].
Reviewer: Robert W. van der Waall (Amsterdam)Uniform convergence criterion for non-harmonic sine series.https://zbmath.org/1460.420052021-06-15T18:09:00+00:00"Oganesyan, K. A."https://zbmath.org/authors/?q=ai:oganesyan.k-aMacroscopic and microscopic anomalous diffusion in comb model with fractional dual-phase-lag model.https://zbmath.org/1460.820222021-06-15T18:09:00+00:00"Liu, Lin"https://zbmath.org/authors/?q=ai:liu.lin"Zheng, Liancun"https://zbmath.org/authors/?q=ai:zheng.liancun"Chen, Yanping"https://zbmath.org/authors/?q=ai:chen.yanping.1Summary: A novel constitutive equation which considers the macroscopic and microscopic relaxation characteristics and the memory and nonlocal characteristics is proposed to describe the anomalous diffusion in comb model. Formulated governing equation with the fractional derivative of order \(1+\alpha\) corresponds to a diffusion-wave one and solutions are obtained analytically with the Laplace and Fourier transforms. As the solutions show, the existence of macroscopic relaxation parameter makes the expression of mean square displacement contain an integral form and the specific value for the microscopic relaxation parameter and macroscopic one changes the coefficient of fractional integral. The particle distribution and mean square displacement of Fick's model and the dual-phase-lag model are same at the short and long time behaviors and the special case of equal macroscopic and microscopic relaxation parameters. The particle distributions and mean square displacement with the effects of different parameters are presented graphically. Results show that the wave characteristic becomes stronger for a larger \(\alpha\), a larger\(\tau_q\) or a smaller \(\tau_P\). For mean square displacement, the magnitude is larger at the short time behavior and smaller at the long time behavior for a smaller \(\alpha\). Besides, for a smaller \(\tau_q\) or a larger \(\tau_P\), the magnitude is larger.Stable Gabor phase retrieval and spectral clustering.https://zbmath.org/1460.940222021-06-15T18:09:00+00:00"Grohs, Philipp"https://zbmath.org/authors/?q=ai:grohs.philipp"Rathmair, Martin"https://zbmath.org/authors/?q=ai:rathmair.martinThis paper concerns the (infinite-dimensional) phase retrieval problem for so-called Gabor systems. Phase retrieval thereby refers to the problem of reconstructing a function \(f\) from the magnitudes of a linear image of it, e.g. \(\vert Af \vert\) for some linear operator \(A\). Since that magnitude is invariant to multiplying \(f\) with a multiplicative complex constant of unit magnitude, the recovery is defined modulus such a constant. More specifically, the publication treats the case of the linear operation being given by the Gabor transform (spectrogram). For a function \(f: \mathbb{R} \to \mathbb{C}\), the Gabor transform is defined as a windowed Fourier transformed with a Gaussian window, i.e. through \(V_\phi(x,y) = \int_\mathbb{R} f(t) e^{-\pi(t-x)^2} e^{-2i\pi y} dt\).
In general, it is not immediate that a function \(f\) can be recovered from the magnitudes \(\vert Af\vert\) -- for the Gabor transform, it however is. On can now ask the question if the recovery is stable, in the sense if \[ \inf_{\vert \theta\vert =1} \Vert{f-\theta g}\Vert_{\mathcal{D}} \leq c(f) \Vert {\vert{V_\phi f}\vert - \vert{V_\phi g}\vert}\Vert_{\mathcal{B}} \] for some (\(f\)-dependent) constant \(c(f)\), and some norms \(\Vert \, \cdot \, \Vert_{\mathcal{D}}\), \(\Vert \, \cdot \, \Vert_{\mathcal{B}}\). It turns out that phase retrieval in infinite dimensions is unstable -- under very natural conditions on \(\mathcal{B}\) and \(\mathcal{D}\), \(\sup_{f} c(f) = \infty\) [\textit{R. Alaifari} et al., Found. Comput. Math. 19, No. 4, 869--900 (2019; Zbl 1440.94010)]. Note that this is a worst-case bound -- indeed, \(c(f)\) may be small for some specific \(f\). To analyze for which \(f\) the recovery is stable is the aim of this paper.
Intuitively, the authors of the paper prove that the stability of the problem gets low only when the measurements \(V_\phi f\) is concentrated on disjoint subsets of \(\mathbb{C}\). The intuition behind the result is that if \(V_\phi\) is concentrated on such `clusters', we can divide \(f\) into two components \(u\) and \(v\), with essentially disjoint support . We can then `flip' one of them (i.e., consider \(g=u-v\)) and obtain a function whose Gabor transform almost has the same magnitude. The authors find a way to formalize this intuition: they prove that the size of the stability constant \(c(f)\) is bounded what they called Cheeger constant of the measure \(\vert V_\phi f\vert \mathrm{d} x \mathrm{d}y\), a notion stemming from the theory of spectral clustering algorithms, which has been well studied.
The proof involves deep theory from several areas of mathematics, such as complex analysis, functional analysis, and spectral Riemannian geometry. The result has huge possible practical implications, as it opens up for designing regularization methods for the phase retrieval problem. Already in this paper, the authors consider an algorithm which subdivides the domain of \(V_\phi f\) into subsets which in themselves have better Cheeger constants, allowing for a more stable retrieval.
Reviewer: Axel Flinth (Göteborg)Polynomial inequalities and Green's functions.https://zbmath.org/1460.260192021-06-15T18:09:00+00:00"Totik, Vilmos"https://zbmath.org/authors/?q=ai:totik.vilmosSummary: The paper discusses some classical polynomial inequalities, their recent extensions to general sets, as well as the potential theory behind them. A unifying feature will be that in many cases the best constant is given in terms of the normal derivative of certain Green's functions.
For the entire collection see [Zbl 1453.41001].A simple shearlet-based 2D Radon inversion with an application to computed tomography.https://zbmath.org/1460.420542021-06-15T18:09:00+00:00"Córdova, Santiago"https://zbmath.org/authors/?q=ai:cordova.santiago"Vera, Daniel"https://zbmath.org/authors/?q=ai:vera.daniel-jSummary: We find a new and simple inversion formula for the 2D Radon transform (RT) with a straight use of the shearlet system and of well-known properties of the RT. Since the continuum theory of shearlets has a natural translation to the discrete theory, we also obtain a computable algorithm that recovers a digital image from noisy samples of the 2D Radon transform which also preserves edges. A very well-known RT inversion in the applied harmonic analysis community is the biorthogonal curvelet decomposition (BCD). The BCD uses an intertwining relation of differential (unbounded) operators between functions in Euclidean and Radon domains. Hence the BCD is ill-posed since the inverse is unstable in the presence of noise. In contrast, our inversion method makes use of an intertwining relation of integral transformations with very smooth kernels and compact support away from the origin in the Fourier domain, i.e. bounded operators. Therefore, we obtain, at least, the same asymptotic behavior of mean-square error as the BCD (and its shearlet version) for the class of cartoon-like functions. Numerical simulations show that our inverse surpasses, by far, the inverse based on the BCD. Our algorithm uses only fast transformations.An approach to the numerical solution of the basic magnetostatic equation for a plane parallel plate with an arbitrarily shaped inclusion.https://zbmath.org/1460.780082021-06-15T18:09:00+00:00"Dyakin, V. V."https://zbmath.org/authors/?q=ai:dyakin.villiam-v"Kudryashova, O. V."https://zbmath.org/authors/?q=ai:kudryashova.o-v"Rayevskii, V. Ya."https://zbmath.org/authors/?q=ai:raevskii.v-yaSummary: A configuration of magnets consisting of a homogeneous plate with a defect in the form of an internal cavity or inclusion of arbitrary shape is considered. Relying on the basic integro-differential equation of magnetostatics for an external field of arbitrary configuration, an expression for the resultant magnetic field strength is obtained in terms of its normal component only on the boundary surface of the defect. An equation for determining this component is derived. A previous work by the authors [``Exact solution of one problem of magnetostatics in bipolar coordinates (continued)'', Russ. J. Nondestr. Test. 52, No. 7, 400--408 (2016; \url{doi:10.1134/S1061830916070032})] is mentioned in which, for a special case of the situation under consideration, a numerical algorithm relying on the indicated equation is implemented and the resultant field components are plotted as functions of physical and geometric parameters of the considered configuration.Stable super-resolution limit and smallest singular value of restricted Fourier matrices.https://zbmath.org/1460.420042021-06-15T18:09:00+00:00"Li, Weilin"https://zbmath.org/authors/?q=ai:li.weilin"Liao, Wenjing"https://zbmath.org/authors/?q=ai:liao.wenjingSummary: We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given \(M+1\) of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance \(\Delta\) between the two closest point sources is less than \(1/M\). We introduce a clumps model where the point sources are closely spaced within several clumps. Under this assumption, we derive a non-asymptotic lower bound for the minimum singular value of a Vandermonde matrix whose nodes are determined by the point sources. Our estimate is given as a weighted \(\ell^2\) sum, where each term only depends on the configuration of each individual clump. The main novelty is that our lower bound obtains an exact dependence on the Super-Resolution Factor \(SRF=(M\Delta)^{-1}\). As noise level increases, the sensitivity of the noise-space correlation function in the MUSIC algorithm degrades according to a power law in SRF where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the sensitivity of MUSIC. We also provide lower and upper bounds for a min-max error of super-resolution for the grid model, which in turn is closely related to the minimum singular value of Vandermonde matrices.A certain studies on Nörlund summability of series.https://zbmath.org/1460.420062021-06-15T18:09:00+00:00"Sahani, S. K."https://zbmath.org/authors/?q=ai:sahani.saroj-kumar|sahani.santosh-kumar"Mishra, L. N."https://zbmath.org/authors/?q=ai:mishra.lakshmi-narayanSummary: In this paper, we have obtained two theorems for Nörlund summability of Fourier series and their conjugate series under very general conditions. These two theorems are closely related to the great works of the analysts \textit{T. Pati} [Indian J. Math. 3, 85--90 (1961; Zbl 0142.31801)], \textit{L. McFadden} [Duke Math. J. 9, 168--207 (1942; Zbl 0061.12106)] and \textit{J. A. Siddiqui} [``On the harmonic summability of Fourier series'', Proc. Indian Acad. Sci., Sect. A, 527--531 (1948)] but not the same.Infinite dimensional compressed sensing from anisotropic measurements and applications to inverse problems in PDE.https://zbmath.org/1460.940182021-06-15T18:09:00+00:00"Alberti, Giovanni S."https://zbmath.org/authors/?q=ai:alberti.giovanni-s"Santacesaria, Matteo"https://zbmath.org/authors/?q=ai:santacesaria.matteoSummary: We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography.Uniqueness of STFT phase retrieval for bandlimited functions.https://zbmath.org/1460.940172021-06-15T18:09:00+00:00"Alaifari, Rima"https://zbmath.org/authors/?q=ai:alaifari.rima"Wellershoff, Matthias"https://zbmath.org/authors/?q=ai:wellershoff.matthiasSummary: We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this note, we demonstrate that by considering signals in Paley-Wiener spaces, it is possible to prove new uniqueness results for STFT phase retrieval. Among those, we establish a first uniqueness theorem for STFT phase retrieval from magnitude-only samples for real-valued signals.