Recent zbMATH articles in MSC 41https://zbmath.org/atom/cc/412022-11-17T18:59:28.764376ZWerkzeugApproximation in the mean on rational curveshttps://zbmath.org/1496.140592022-11-17T18:59:28.764376Z"Biswas, Shibananda"https://zbmath.org/authors/?q=ai:biswas.shibananda"Putinar, Mihai"https://zbmath.org/authors/?q=ai:putinar.mihaiFor a positive Borel measure \(\mu\), compactly supported on the complex plane and without point masses, a theorem of Thomson, subsequently generalized to rational functions by Brennan, states that the closure \(P^2(\mu)\) in \(L^2(\mu)\) of the polynomials in one complex variable is different from \(L^2(\mu)\) if and only if there exist analytic bounded point evaluations. Now, given a complex affine curve \(\mathcal{V} \in \mathbb{C}^n\) and a positive Borel measure \(\mu\) supported by a compact subset \(K\) of \(\mathcal{V}\), the goal of the paper is to establish conditions that ensure the validity of Thomson's Theorem on algebraic curves \(\mathcal{V}\), thus relating the density of polynomials in Lebesgue \(L^2\)-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan on rational curves are also obtained.
Reviewer: Carlos Hermoso Ortíz (Madrid)On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functionshttps://zbmath.org/1496.260412022-11-17T18:59:28.764376Z"You, Minghui"https://zbmath.org/authors/?q=ai:you.minghui"Sun, Xia"https://zbmath.org/authors/?q=ai:sun.xia"Fan, Xiansheng"https://zbmath.org/authors/?q=ai:fan.xiansheng(no abstract)On the \(L^2\)-norm of Gegenbauer polynomialshttps://zbmath.org/1496.330092022-11-17T18:59:28.764376Z"Ferizović, Damir"https://zbmath.org/authors/?q=ai:ferizovic.damir|ferizovic.damir.1Summary: Gegenbauer, also known as ultra-spherical, polynomials appear often in numerical analysis or interpolation. In the present text we find a recursive formula for and compute the asymptotic behavior of their \(L^2\)-norm.Analytic solution of the SEIR epidemic model via asymptotic approximanthttps://zbmath.org/1496.340872022-11-17T18:59:28.764376Z"Weinstein, Steven J."https://zbmath.org/authors/?q=ai:weinstein.steven-j"Holland, Morgan S."https://zbmath.org/authors/?q=ai:holland.morgan-s"Rogers, Kelly E."https://zbmath.org/authors/?q=ai:rogers.kelly-e"Barlow, Nathaniel S."https://zbmath.org/authors/?q=ai:barlow.nathaniel-sSummary: An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in \(\ln S\) and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant [\textit{N. S. Barlow} et al., Classical Quantum Gravity 34, No. 13, Article ID 135017, 16 p. (2017; Zbl 1367.83019)] in the form of a modified symmetric Padé approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.Steklov eigenvalues of nearly spherical domainshttps://zbmath.org/1496.352622022-11-17T18:59:28.764376Z"Viator, Robert"https://zbmath.org/authors/?q=ai:viator.robert-jun"Osting, Braxton"https://zbmath.org/authors/?q=ai:osting.braxtonThe authors consider the Steklov eigenproblem on a bounded domain \(\Omega \subset \mathbb{R}^d\):
\[
\Delta u= 0 \quad \text{in}\quad \Omega,\qquad \partial_n u = \lambda u \quad\text{on}\quad \partial \Omega.
\]
The eigenvalues are enumerated in increasing order: \(0 = \lambda_0 (\Omega) < \lambda_1(\Omega) \leq \lambda_2(\Omega) \dots\).
The first main result concerns a nearly-spherical domain \(\Omega = \Omega_\varepsilon\), the boundary of which is written in terms of the spherical coordinates (radius \(r\), inclination \(\theta\in[0,\pi]\), azimuth \(\phi\in[0,2\pi]\)) and is expanded in the basis of real spherical harmonics,
\[
\Omega_\varepsilon=\{(r,\theta,\phi)\colon 0\leqslant r\leqslant 1 + \varepsilon \rho(\theta,\phi)\},\qquad \rho(\theta,\phi)=\sum_{\ell=0}^{\infty}\sum_{m = -\ell}^\ell A_{\ell,m} Y_{\ell,m}(\theta,\phi)
\]
where \(\rho\) is a given \(C^1(\partial \Omega_0)\) perturbation function. As \(\varepsilon = 0\), the domain \(\Omega_0\) is the unit ball and the eigenvalues are \(\lambda_{\ell,m} = \ell\) (multiplicity \(2\ell +1\)) with corresponding eigenfunctions
\[
u_{\ell,m} (r,\theta,\phi) = r^\ell Y_{\ell,m}(\theta,\phi), \qquad \qquad \ell \in \mathbb{N}, \ |m|\leqslant \ell.
\]
The first main results reads as follows. Fix \(k \in \mathbb{N}\). The Steklov eigenvalues, \(\lambda_n(\varepsilon)\), for \(n \in \{ k^2, \ \ldots, \ (k+1)^2 - 1 \}\) consist of at most \(2k+1\) branches of analytic functions which have at most algebraic singularities near \(\varepsilon=0\). At first-order in \(\varepsilon\), the perturbation is given by the eigenvalues of the symmetric matrix \(M^{(k)}\):
\[
\lambda_{n}(\varepsilon) = k + \varepsilon \lambda_{n}^{(1)} + O (\varepsilon^2),
\]
where \(\lambda_{n}(\varepsilon)\) the eigenvalues of a real, symmetric \(2k+1 \times 2k+1\) matrix, denoted \(M^{(k)}\), whose entries are given by
\[
M^{(k)}_{m,n} = - \frac{1}{2} \sum_{p=0}^{\infty}\sum_{q = -p}^p A_{p,q} \left( p(p+1) + 2k \right) \iint \limits_{S^2} Y_{p,q}(\theta,\phi) Y_{k,m}(\theta,\phi) Y_{k,n}(\theta,\phi) \ dS .
\]
Denote the volume-normalized Steklov eigenvalue by \(\Lambda_\ell(\Omega ) := \lambda_\ell(\Omega) \cdot |\Omega|^{\frac 1 3}\). The second result is as follows. Let \(k \in \mathbb{N}\). Then \(\Lambda_{k^2}\) is stationary for a ball in the sense that, for every perturbation function \(\rho\), the map \(\varepsilon \mapsto \Lambda_{k^2}(\Omega_\varepsilon)\) is non-increasing in \(|\varepsilon|\) for \(|\varepsilon|\) sufficiently small. The eigenvalue \(\Lambda_{(k+1)^2-1}\) is not maximized by the ball.
Reviewer: Denis Borisov (Ufa)Defocusing NLS equation with nonzero background: large-time asymptotics in a solitonless regionhttps://zbmath.org/1496.353712022-11-17T18:59:28.764376Z"Wang, Zhaoyu"https://zbmath.org/authors/?q=ai:wang.zhaoyu"Fan, Engui"https://zbmath.org/authors/?q=ai:fan.enguiSummary: We consider the Cauchy problem for the defocusing Schrödinger (NLS) equation with a nonzero background
\[
\begin{aligned}
&i q_t + q_{x x} - 2(| q |^2 - 1) q = 0,\\
&q(x, 0) = q_0(x), \quad \lim_{x \to \pm \infty} q_0(x) = \pm 1.
\end{aligned}
\]
Recently, for the space-time region \(| x /(2 t) | < 1\) which is a solitonic region without stationary phase points on the jump contour, \textit{S. Cuccagna} and \textit{R. Jenkins} [Commun. Math. Phys. 343, No. 3, 921--969 (2016; Zbl 1342.35326)] presented the asymptotic stability of the \(N\)-soliton solutions for the NLS equation by using the \(\overline{\partial}\) generalization of the Deift-Zhou nonlinear steepest descent method. Their large-time asymptotic expansion takes the form
\[
q (x, t) = T (\infty)^{- 2} q^{s o l, N} (x, t) + \mathcal{O} (t^{- 1})
\tag{0.1}
\]
whose leading term is N-soliton and the second term \(\mathcal{O}(t^{- 1})\) is a residual error from a \(\overline{\partial}\)-equation. In this paper, we are interested in the large-time asymptotics in the space-time region \(| x /(2 t) | > 1\) which is outside the soliton region, but there will be two stationary points appearing on the jump contour \(\mathbb{R}\). We found an asymptotic expansion that is different from (0.1)
\[
q (x, t) = e^{- i \alpha (\infty)} (1 + t^{- 1 / 2} h (x, t)) + \mathcal{O} (t^{- 3 / 4}),
\tag{0.2}
\]
whose leading term is a nonzero background, the second \(t^{- 1 / 2}\) order term is from the continuous spectrum and the third term \(\mathcal{O}(t^{- 3 / 4})\) is a residual error from a \(\overline{\partial}\)-equation. The above two asymptotic results (0.1) and (0.2) imply that the region \(| x /(2 t) | < 1\) considered by Cuccagna and Jenkins [loc. cit.] is a fast decaying soliton solution region, while the region \(| x /(2 t) | > 1\) considered by us is a slow decaying nonzero background region.Exit versus escape for stochastic dynamical systems and application to the computation of the bursting time duration in neuronal networkshttps://zbmath.org/1496.370822022-11-17T18:59:28.764376Z"Zonca, Lou"https://zbmath.org/authors/?q=ai:zonca.lou"Holcman, David"https://zbmath.org/authors/?q=ai:holcman.davidSummary: We study the exit time of two-dimensional dynamical systems perturbed by a small noise that exhibits two peculiar behaviors: (1) The maximum of the probability density function of trajectories is not located at the point attractor. The distance between the maximum and the attractor increases with the noise amplitude \(\sigma\), as shown by using WKB approximation and numerical simulations. (2) For such systems, exiting from the basin of attraction is not sufficient to guarantee a full escape, due to trajectories that can return several times inside the basin of attraction after crossing the boundary, before eventually escaping far away. We apply these results to study neuronal networks that can generate bursting events. To analyze interburst durations and their statistics, we study the phase space of a mean-field model, based on synaptic short-term changes, that exhibit burst and interburst dynamics. We find that the interburst corresponds to an escape with multiple reentries inside the basin of attraction. To conclude, escaping far away from a basin of attraction is not equivalent to reaching the boundary, thus providing an explanation for non-Poissonian long interburst durations present in neuronal dynamics.A new generalized version of Korovkin-type approximation theoremhttps://zbmath.org/1496.400132022-11-17T18:59:28.764376Z"Khan, Vakeel A."https://zbmath.org/authors/?q=ai:khan.vakeel-a"Khan, Izhar Ali"https://zbmath.org/authors/?q=ai:khan.izhar-ali"Hazarika, Bipan"https://zbmath.org/authors/?q=ai:hazarika.bipanIn statistical convergence, the convergence condition is obtained just for a majority of elements. Therefore it extends the concept of ordinary convergence and it is an effective tool to obtain strong results. Recently, there have been obtained many generalizations of statistical convergence by combining it with ideal, measure and mean.
In this paper, the authors introduce \(\mu\)-statistical measurable convergence, ideal \(\mu\)-statistical measurable convergence, \(\mu\)-deferred ideal statistical measurable convergence, \(\mu\)-deferred ideal statistical mean convergence of order \(\alpha\), \(\mu\)-deferred ideal statistical mean convergence in measure of order \(\alpha\), \(\mu\)-deferred statistical Lebesgue measurable convergence under integral. They examine the implications between these concepts of convergences in detail. They also provide examples for the converse parts. At the end, as an application they present Korovkin-type results with the use of these notions.
Reviewer: Tuğba Yurdakadim (Bilecik)A class of Birkhoff type interpolation and applicationshttps://zbmath.org/1496.410012022-11-17T18:59:28.764376Z"Mahmoodi, A."https://zbmath.org/authors/?q=ai:mahmoodi.amin|mahmoodi.ali|mahmoodi.akram|mahmoodi.alireza"Nazarzadeh, A."https://zbmath.org/authors/?q=ai:nazarzadeh.aSummary: In this paper, a class of Birkhoff type interpolation problem on arbitrary knots is studied. This class of Birkhoff interpolation is a generalization of a Birkhoff problem, which is included in one of the articles in the referenced list. Because the behavior of first and second order derivatives are more important than the higher order, therefore, for a basic function, a better approximation will be obtained than higher order derivatives. The low error value calculated for some functions using this class represents the relative importance of this class of Birkhoff interpolation problem. One of the uses of this class from the Birkhoff polynomials is to review and propose a quadrature formula for this class. One of the ways to reduce the error for interpolation polynomials of this class of Birkhoff interpolation problem is to select the Chebyshev polynomial zeros as nodal points, which can be used to solve numerical differential equations.Samples of homogeneous functionshttps://zbmath.org/1496.410022022-11-17T18:59:28.764376Z"Dăianu, Dan M."https://zbmath.org/authors/?q=ai:daianu.dan-mSummary: We present an algorithm for extracting samples of homogeneity from functions between sets endowed with actions. In this way, we extend Taylor's formula with the remainder of Peano type in a very wide framework. We illustrate the versatility of this procedure by giving approximations with polynomials of homogeneous functions for some non-differentiable functions.Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variationhttps://zbmath.org/1496.410032022-11-17T18:59:28.764376Z"Liu, Wenjie"https://zbmath.org/authors/?q=ai:liu.wen-jie"Wang, Li-Lian"https://zbmath.org/authors/?q=ai:wang.lilian"Wu, Boying"https://zbmath.org/authors/?q=ai:wu.boyingA new fractional Taylor formula is obtained for singular functions whose Caputo fractional derivatives are of bounded variation. This enables to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions.The optimal (weighted) \(L^{\infty}\)-estimates and \(L^2\)-estimates of the Legendre polynomial approximations are also derived.
Reviewer: Zoltán Finta (Cluj-Napoca)Notes of a numerical analyst. What's the degree of \(x^n\)?https://zbmath.org/1496.410042022-11-17T18:59:28.764376Z"Trefethen, Nick"https://zbmath.org/authors/?q=ai:trefethen.lloyd-n(no abstract)Quantitative approximation by Kantorovich-Choquet quasi-interpolation neural network operatorshttps://zbmath.org/1496.410052022-11-17T18:59:28.764376Z"Anastassiou, George A."https://zbmath.org/authors/?q=ai:anastassiou.george-aSummary: In this article we present univariate and multivariate basic approximation by Kantorovich-Choquet type quasi-interpolation neural network operators with respect to supremum norm. This is done with rates using the first univariate and multivariate moduli of continuity. We approximate continuous and bounded functions on \(\mathbb R^N\), \(N \subset \mathbb N\). When they are also uniformly continuous we have pointwise and uniform convergences.\(L^p\)-Bernstein inequalities on \(C^2\)-domains and applications to discretizationhttps://zbmath.org/1496.410062022-11-17T18:59:28.764376Z"Dai, Feng"https://zbmath.org/authors/?q=ai:dai.feng"Prymak, Andriy"https://zbmath.org/authors/?q=ai:prymak.andriy-vSummary: We prove a new Bernstein type inequality in \(L^p\) spaces associated with the normal and the tangential derivatives on the boundary of a general compact \(C^2\)-domain. We give two applications: Marcinkiewicz type inequality for discretization of \(L^p\) norms and positive cubature formulas. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.Design of accurate formulas for approximating functions in weighted Hardy spaces by discrete energy minimizationhttps://zbmath.org/1496.410072022-11-17T18:59:28.764376Z"Tanaka, Ken'ichiro"https://zbmath.org/authors/?q=ai:tanaka.kenichiro"Sugihara, Masaaki"https://zbmath.org/authors/?q=ai:sugihara.masaakiSummary: We propose a simple and effective method for designing approximation formulas for weighted analytic functions. We consider spaces of such functions according to weight functions expressing the decay properties of the functions. Then we adopt the minimum worst error of the \(n\)-point approximation formulas in each space for characterizing the optimal sampling points for the approximation. In order to obtain approximately optimal sampling points we consider minimization of a discrete energy related to the minimum worst error. Consequently, we obtain an approximation formula and its theoretical error estimate in each space. In addition, from some numerical experiments, we observe that the formula generated by the proposed method outperforms the corresponding formula derived with sinc approximation, which is near optimal in each space.Approximation of discontinuous signals by exponential sampling serieshttps://zbmath.org/1496.410082022-11-17T18:59:28.764376Z"Angamuthu, Sathish Kumar"https://zbmath.org/authors/?q=ai:angamuthu.sathish-kumar"Kumar, Prashant"https://zbmath.org/authors/?q=ai:kumar.prashant.1|kumar.prashant"Ponnaian, Devaraj"https://zbmath.org/authors/?q=ai:ponnaian.devarajThe approximation of jump discontinuity functions \(f\) by exponential sampling series \(S_{w}^{\chi}f\) is established using a representation lemma for \(S_{w}^{\chi}f\). The rate of approximation of \(S_{w}^{\chi}f\) is obtained in terms of logarithmic modulus of continuity, the round-off and time-jitter errors are also studied. The construction of a family of Mellin band-limited kernels is given such that \(\chi(1)=0\) for which \(S_{w}^{\chi}f\) converges at any jump discontinuities. Finally some graphical representation of approximation of discontinuous functions by \(S_{w}^{\chi}f\) are obtained.
Reviewer: Zoltán Finta (Cluj-Napoca)Some properties of Kantorovich variant of Chlodowsky-Szász operators induced by Boas-Buck type polynomialshttps://zbmath.org/1496.410092022-11-17T18:59:28.764376Z"Braha, Naim L."https://zbmath.org/authors/?q=ai:braha.naim-latif"Loku, Valdete"https://zbmath.org/authors/?q=ai:loku.valdete"Mansour, Toufik"https://zbmath.org/authors/?q=ai:mansour.toufikIn this paper the authors have introduced the Kantorovich variant of Chlodowsky-Szász operators inspred by Boas-Buck type polynomials which is introduced by \textit{I. Chlodovsky} [Compos. Math. 4, 380--393 (1937; Zbl 0016.35401)], and have examined some properties of the new operators. By using modulus of continuity and Peetre's K-functional, some results about rate of convergence have been given for the operators. Also Voronovskaya type result the operators has been stated. On the other hand weighted versions of the above results have been examined.
Reviewer: Emre Taş (Kırşehir)Approximation by max-product operators of Kantorovich typehttps://zbmath.org/1496.410102022-11-17T18:59:28.764376Z"Coroianu, Lucian"https://zbmath.org/authors/?q=ai:coroianu.lucian-c"Gal, Sorin G."https://zbmath.org/authors/?q=ai:gal.sorin-gheorgheSummary: The main goal of this survey is to describe the results of the present authors concerning approximation properties of various max-product Kantorovich operators, fulfilling thus this gap in their very recent research monograph [\textit{B. Bede} et al., Approximation by max-product type operators. Cham: Springer (2016; Zbl 1358.41013)]. Section 1 contains a short introduction in the topic. In Sect. 2, after presenting some general results, we state approximation results including upper estimates, direct and inverse results, localization results and shape preserving results, for the max-product: Bernstein-Kantorovich operators, truncated and non-truncated Favard-Szász-Mirakjan-Kantorovich operators, truncated and non-truncated Baskakov-Kantorovich operators, Meyer-König-Zeller-Kantorovich operators, Hermite-Fejér-Kantorovich operators based on the Chebyshev knots of first kind, discrete Picard-Kantorovich operators, discrete Weierstrass-Kantorovich operators and discrete Poisson-Cauchy-Kantorovich operators. All these approximation properties are deduced directly from the approximation properties of their corresponding usual max-product operators. Section 3 presents the approximation properties with quantitative estimates in the \(L^p\)-norm, \(1 \leq p \leq +\infty\), for the Kantorovich variant of the truncated max-product sampling operators based on the Fejér kernel. In Sect. 4, we introduce and study the approximation properties in \(L^p\)-spaces, \(1 \leq p \leq +\infty\) for truncated max-product Kantorovich operators based on generalized type kernels depending on two functions \(\phi\) and \(\psi\) satisfying a set of suitable conditions. The goal of Sect. 5 is to present approximation in \(L^p\), \(1 \leq p \leq +\infty\), by sampling max-product Kantorovich operators based on generalized kernels, not necessarily with bounded support, or generated by sigmoidal functions. Several types of kernels for which the theory applies and possible extensions and applications to higher dimensions are presented. Finally, some new directions for future researches are presented, including applications to learning theory.
For the entire collection see [Zbl 1483.00042].Approximation by mixed operators of max-product-Choquet typehttps://zbmath.org/1496.410112022-11-17T18:59:28.764376Z"Gal, Sorin G."https://zbmath.org/authors/?q=ai:gal.sorin-gheorghe"Iancu, Ionut T."https://zbmath.org/authors/?q=ai:iancu.ionut-tSummary: The main aim of this chapter is to introduce several mixed operators between Choquet integral operators and max-product operators and to study their approximation, shape preserving, and localization properties. Section 2 contains some preliminaries on the Choquet integral. In Sect. 3, we obtain quantitative estimates in uniform and pointwise approximation for the following mixed type operators: max-product Bernstein-Kantorovich-Choquet operator, max-product Szász-Mirakjan-Kantorovich-Choquet operators, nontruncated and truncated cases, and max-product Baskakov-Kantorovich-Choquet operators, nontruncated and truncated cases. We show that for large classes of functions, the max-product Bernstein-Kantorovich-Choquet operators approximate better than their classical correspondents, and we construct new max-product Szász-Mirakjan-Kantorovich-Choquet and max-product Baskakov-Kantorovich-Choquet operators, which approximate uniformly \(f\) in each compact subinterval of \([0, +\infty)\) with the order \(\omega_1(f; \sqrt{\lambda_n})\), where \(\lambda_n \searrow 0\) arbitrary fast. Also, shape preserving and localization results for max-product Bernstein-Kantorovich-Choquet operators are obtained. Section 4 contains quantitative approximation results for discrete max-product Picard-Kantorovich-Choquet, discrete max-product Gauss-Weierstrass-Kantorovich-Choquet operators, and discrete max-product Poisson-Cauchy-Kantorovich-Choquet operators. Section 5 deals with the approximation properties of the max-product Kantorovich-Choquet operators based on \((\varphi, \psi)\)-kernels. It is worth to mention that with respect to their max-product correspondents, while they keep their good properties, the mixed max-product Choquet operators present, in addition, the advantage of a great flexibility by the many possible choices for the families of set functions used in their definitions. The results obtained present potential applications in sampling theory, neural networks, and learning theory.
For the entire collection see [Zbl 1485.65002].Approximation by linear combinations of translates of a single functionhttps://zbmath.org/1496.410122022-11-17T18:59:28.764376Z"Dũng, Dinh"https://zbmath.org/authors/?q=ai:dinh-dung."Huy, Vu Nhat"https://zbmath.org/authors/?q=ai:huy.vu-nhatSummary: We study approximation of periodic functions by arbitrary linear combinations of n translates of a single function. We construct some linear methods of this approximation for univariate functions in the class induced by the convolution with a single function, and prove upper bounds of the \(L^p\)-approximation convergence rate by these methods, when \(n\rightarrow\infty\), for \(1\leq p\leq\infty\). We also generalize these results to classes of multivariate functions defined as the convolution with the tensor product of a single function. In the case \(p=2\), for this class, we also prove a lower bound of the quantity characterizing best approximation of by arbitrary linear combinations of \(n\) translates of arbitrary function.Factorizations of bivariate Taylor series via inverse power productshttps://zbmath.org/1496.410132022-11-17T18:59:28.764376Z"Elewoday, Mohamed"https://zbmath.org/authors/?q=ai:elewoday.mohamed"Gingold, Harry"https://zbmath.org/authors/?q=ai:gingold.harry"Quaintance, Jocelyn"https://zbmath.org/authors/?q=ai:quaintance.jocelynIn this paper, the authors convert \(f(x, y)\) into the formal product \(\Pi^{\infty} _{\substack{p=1 \\ m+n=p}}(1-h_{m,n}x^{m}y^{n})^{ -1}\), namely the inverse power product expansion in two independent variables and also discuss various combinatorial interpretations provided by these inverse power product expansions.
English is clear in style and consistent with the standards of usage.
Reviewer: Aida Tagiyeva (Baku)Simple accurate balanced asymptotic approximation of Wallis' ratio using Euler-Boole alternating summationhttps://zbmath.org/1496.410142022-11-17T18:59:28.764376Z"Lampret, Vito"https://zbmath.org/authors/?q=ai:lampret.vitoSummary: For integers \(m\geq 1\) and \(q\geq 2\), the Wallis ratio \(w_m:=\prod\limits_{k=1}^m\frac{2k-1}{2k}\) is estimated as
\[
\bigg|w_m-\frac{1}{\sqrt{m\pi}}\exp\bigg(-\sum\limits_{i=1}^{\lfloor q/2\rfloor}\frac{(1-4^{-1})B_{2i}}{i(2i-1)\cdot m^{2i-1}}\bigg)\bigg|<\frac{1}{2}\exp(\rho_q^\ast(m))\cdot\rho^\ast_q(m),
\]
where \(B_k\) are the Bernoulli coefficient and
\[
|\rho_q^\ast(m)|<\frac{\pi(q-2)!}{3(2m\pi)^{q-1}}<\frac{\pi}{3}\sqrt{\frac{2\pi}{q-1}}\cdot\left(\frac{q-1}{2me\pi}\right)^{q-1}\exp\left(\frac{1}{12(q-q)}\right).
\]
Some accurate asymptotic estimates of \(\pi\) in terms of \(w_m\) are also given.On optimal recovery of values of linear operators from information known with a stochastic errorhttps://zbmath.org/1496.410152022-11-17T18:59:28.764376Z"Krivosheev, Kirill Yu."https://zbmath.org/authors/?q=ai:krivosheev.kirill-yuApproximation by quasi-interpolation operators and Smolyak's algorithmhttps://zbmath.org/1496.420012022-11-17T18:59:28.764376Z"Kolomoitsev, Yurii"https://zbmath.org/authors/?q=ai:kolomoitsev.yurii-sSummary: We study approximation of multivariate periodic functions from Besov and Triebel-Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the \(L_q\)-norm for functions from the Besov spaces \(\text{B}_{p,\theta}^s(\mathbb{T}^d)\) and the Triebel-Lizorkin spaces \(\text{F}_{p,\theta}^s(\mathbb{T}^d)\) for all \(s>0\) and admissible \(1 \leq p\), \(\theta\leq\infty\) as well as provide analogues of the Littlewood-Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuityhttps://zbmath.org/1496.420322022-11-17T18:59:28.764376Z"Fedunyk-Yaremchuk, O. V."https://zbmath.org/authors/?q=ai:fedunyk-yaremchuk.oksana-volodymyrivna"Hembars'ka, S. B."https://zbmath.org/authors/?q=ai:hembarska.svitlana-borysivnaSummary: In this paper, we continue the study of approximation characteristics of the classes \(B^{\Omega}_{p,\theta}\) of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes \(B^{\Omega}_{p,\theta}\) in the space \(L_q\), \(1\leq p<q<\infty\), and also establish the exact-order estimates of approximation for these classes of functions in the space \(L_q\) by using linear operators satisfying certain conditions.The metric projections onto closed convex cones in a Hilbert spacehttps://zbmath.org/1496.520062022-11-17T18:59:28.764376Z"Qiu, Yanqi"https://zbmath.org/authors/?q=ai:qiu.yanqi"Wang, Zipeng"https://zbmath.org/authors/?q=ai:wang.zipengSummary: We study the metric projection onto the closed convex cone in a real Hilbert space \(\mathscr{H}\) generated by a sequence \(\mathcal{V} = \{v_n\}_{n=0}^\infty\). The first main result of this article provides a sufficient condition under which the closed convex cone generated by \(\mathcal{V}\) coincides with the following set:
\[
\mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0, \text{ the series }\sum_{n=0}^\infty a_n v_n \text{ converges in } \mathscr{H}\bigg\}.
\]
Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto \(\mathcal{C}[[\mathcal{V}]]\). As an application, we obtain the best approximations of many concrete functions in \(L^2([-1,1])\) by polynomials with nonnegative coefficients.On the weak laws of large numbers for compound random sums of independent random variables with convergence rateshttps://zbmath.org/1496.600222022-11-17T18:59:28.764376Z"Tran, Loc Hung"https://zbmath.org/authors/?q=ai:hung.tran-locSummary: The weak laws of large numbers are extremely intuitive and applicable results in various fields of probability theory and mathematical statistics. Compound random sums are extensions of classical random sums where the random number of summands is a partial sum of independent and identically distributed positive integer-valued random variables, assuming independence of summands. In this paper, a weak laws of large numbers for normalized compound random sums of independent (not necessarily identically distributed) random variables is studied and the convergence rates in types of ``Small-o'' and ``Large-\(\mathcal{O}\)'' error estimates are established, using Trotter's distance approach. The obtained results in this paper are extensions and generalizations of the known classical ones.Inverse central ordering for the Newton interpolation formulahttps://zbmath.org/1496.650132022-11-17T18:59:28.764376Z"Carnicer, J. M."https://zbmath.org/authors/?q=ai:carnicer.jesus-miguel"Khiar, Y."https://zbmath.org/authors/?q=ai:khiar.yasmina"Peña, J. M."https://zbmath.org/authors/?q=ai:pena.juan-manuelSummary: An inverse central ordering of the nodes is proposed for the Newton interpolation formula. This ordering may improve the stability for certain distributions of nodes. For equidistant nodes, an upper bound of the conditioning is provided. This bound is close to the bound of the conditioning in the Lagrange interpolation formula, whose conditioning is the lowest. This ordering is related to a pivoting strategy of a matrix elimination procedure called Neville elimination. The results are illustrated with examples.The discrete analogue of the differential operator \( \frac{\mathrm{d}^{2m}}{\mathrm{d}\,x^{2m}}+2\omega^2\frac{\mathrm{d}^{2m-2}}{\mathrm{d}\,x^{2m-2}}+\omega^4\frac{\mathrm{d}^{2m-4}}{\mathrm{d}\,x^{2m-4}} \)https://zbmath.org/1496.650152022-11-17T18:59:28.764376Z"Hayotov, A. R."https://zbmath.org/authors/?q=ai:hayotov.abdullo-rakhmonovich(no abstract)Singular value decomposition versus sparse grids: refined complexity estimateshttps://zbmath.org/1496.650202022-11-17T18:59:28.764376Z"Griebel, Michael"https://zbmath.org/authors/?q=ai:griebel.michael"Harbrecht, Helmut"https://zbmath.org/authors/?q=ai:harbrecht.helmutSummary: We compare the cost complexities of two approximation schemes for functions that live on the product domain \(\varOmega _1\times \varOmega _2\) of sufficiently smooth domains \(\varOmega _1\subset \mathbb{R}^{n_1}\) and \(\varOmega _2\subset \mathbb{R}^{n_2} \), namely the singular value/Karhunen-Lòeve decomposition and the sparse grid representation. We assume that appropriate finite element methods with associated orders \(r_1\) and \(r_2\) of accuracy are given on the domains \(\varOmega_1\) and \(\varOmega_2\), respectively. This setting reflects practical needs, since often black-box solvers are used in numerical simulation, which restrict the freedom in the choice of the underlying discretization. We compare the cost complexities of the associated singular value decomposition and the associated sparse grid approximation. It turns out that, in this situation, the approximation by the sparse grid is always equal or superior to the approximation by the singular value decomposition. The results in this article improve and generalize those from the study by the authors [ibid. 34, No. 1, 28--54 (2014; Zbl 1287.65009)]. Especially, we consider the approximation of functions from generalized isotropic \textit{and} anisotropic Sobolev spaces.Degree of multivariate approximation by superposition of a sigmoidal functionhttps://zbmath.org/1496.650212022-11-17T18:59:28.764376Z"Hahm, N. W."https://zbmath.org/authors/?q=ai:hahm.n-wSummary: Multivariate approximation by superposition of a sigmoidal function has been investigated by many authors. \textit{G. Cybenko} [Math. Control Signals Syst. 2, No. 4, 303--314 (1989; Zbl 0679.94019)] suggested a non-constructive proof of multivariate approximation and \textit{N. W. Hahm} [J. Anal. Appl. 18, No. 2, 119--131 (2020; Zbl 1466.65018)] showed the density result of multivariate approximation using a constructive proof.
In this paper, we examine the complexity result of multivariate approximation by superposition of a sigmoidal function and suggest an approximation order using the modulus of continuity. Our proofs are constructive.In reference to a self-referential approach towards smooth multivariate approximationhttps://zbmath.org/1496.650222022-11-17T18:59:28.764376Z"Pandey, K. K."https://zbmath.org/authors/?q=ai:pandey.kshitij-kumar"Viswanathan, P."https://zbmath.org/authors/?q=ai:viswanathan.puthan-veeduSummary: Approximation of a multivariate function is an important theme in the field of numerical analysis and its applications, which continues to receive a constant attention. In this paper, we provide a parameterized family of self-referential (fractal) approximants for a given multivariate smooth function defined on an axis-aligned hyper-rectangle. Each element of this class preserves the smoothness of the original function and interpolates the original function at a prefixed gridded data set. As an application of this construction, we deduce a fractal methodology to approach a multivariate Hermite interpolation problem. This part of our paper extends the classical bivariate Hermite's interpolation formula by \textit{A. C. Ahlin} [Math. Comput. 18, 264--273 (1964; Zbl 0122.12501)] in a twofold sense: (i) records, in particular, a multivariate generalization of this bivariate interpolation theory; (ii) replaces the unicity of the Hermite interpolant with a parameterized family of self-referential Hermite interpolants which contains the multivariate analogue of Ahlin's interpolant as a particular case. Some related aspects including the approximation by multivariate self-referential functions preserving Popoviciu convexity are given, too.Fast and accurate evaluation of dual Bernstein polynomialshttps://zbmath.org/1496.650352022-11-17T18:59:28.764376Z"Chudy, Filip"https://zbmath.org/authors/?q=ai:chudy.filip"Woźny, Paweł"https://zbmath.org/authors/?q=ai:wozny.pawelSummary: Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis, and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of these polynomials and their linear combinations. New simple recurrence relations of low order satisfied by dual Bernstein polynomials are given. In particular, a first-order non-homogeneous recurrence relation linking dual Bernstein and shifted Jacobi orthogonal polynomials has been obtained. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all \(n+1\) dual Bernstein polynomials of degree \(n\) with \(O(n)\) computational complexity.Numerical differentiation on scattered data through multivariate polynomial interpolationhttps://zbmath.org/1496.650362022-11-17T18:59:28.764376Z"Dell'Accio, Francesco"https://zbmath.org/authors/?q=ai:dellaccio.francesco"Di Tommaso, Filomena"https://zbmath.org/authors/?q=ai:di-tommaso.filomena"Siar, Najoua"https://zbmath.org/authors/?q=ai:siar.najoua"Vianello, Marco"https://zbmath.org/authors/?q=ai:vianello.marcoSummary: We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.A multivariate version of Hammer's inequality and its consequences in numerical integrationhttps://zbmath.org/1496.650392022-11-17T18:59:28.764376Z"Guessab, Allal"https://zbmath.org/authors/?q=ai:guessab.allal"Semisalov, Boris"https://zbmath.org/authors/?q=ai:semisalov.boris-vladimirovichSummary: According to Hammer's inequality [\textit{P. C. Hammer}, Math. Mag. 31, 193--195 (1958; Zbl 0085.11402)], which is a refined version of the famous Hermite-Hadamard inequality, the midpoint rule is always more accurate than the trapezoidal rule for any convex function defined on some real numbers interval \([a,b]\). In this paper, we consider some properties of a multivariate extension of this result to an arbitrary convex polytope. The proof is based on the use of Green formula. In doing so, we will prove an inequality recently conjectured in [the authors, BIT 58, No. 3, 613--660 (2018; Zbl 1496.65040)] about a natural multivariate version of the classical trapezoidal rule. Our proof is based on a generalization of Hammer's inequality in a multivariate setting. It also provides a way to construct new ``extended'' cubature formulas, which give a reasonably good approximation to integrals in which they have been tested. We particularly pay attention to the explicit expressions of the best possible constants appearing in the error estimates for these cubature formulas.Numerical integration using integrals over hyperplane sections of simplices in a triangulation of a polytopehttps://zbmath.org/1496.650402022-11-17T18:59:28.764376Z"Guessab, Allal"https://zbmath.org/authors/?q=ai:guessab.allal"Semisalov, Boris"https://zbmath.org/authors/?q=ai:semisalov.boris-vladimirovichSummary: In this paper, we consider the problem of approximating a definite integral of a given function \(f\) when, rather than its values at some points, a number of integrals of \(f\) over some hyperplane sections of simplices in a triangulation of a polytope \(P\) in \(\mathbb{R}^d\) are only available. We present several new families of ``extended'' integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson's rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.Optimal quadrature formulas with polynomial weight in the space \(L^{(m)}_2(0,1)\)https://zbmath.org/1496.650412022-11-17T18:59:28.764376Z"Ismoilov, S. I."https://zbmath.org/authors/?q=ai:ismoilov.s-i(no abstract)Fibonacci polynomials for the numerical solution of variable-order space-time fractional Burgers-Huxley equationhttps://zbmath.org/1496.651792022-11-17T18:59:28.764376Z"Heydari, M. H."https://zbmath.org/authors/?q=ai:heydari.mohammad-hossien"Avazzadeh, Z."https://zbmath.org/authors/?q=ai:avazzadeh.zakiehSummary: In this article, the variable-order (VO) space-time fractional version of the Burgers-Huxley equation is introduced with fractional differential operator of the Caputo type. The collocation technique based on the Fibonacci polynomials (FPs) is developed for finding the approximate solution of this equation. In order to implement the presented method, some novel operational matrices of derivative (including ordinary and fractional derivatives) are extracted for the FPs. Moreover, the roots of the Chebyshev polynomials of the first kind are chosen as the collocation points which reduce the equation to a system of algebraic equations more efficiency. Ultimately, we obtain the solution of the VO space-time fractional Burgers-Huxley equation in terms of the FPs. The devised method is validated by finding an error bound for the truncated series of the Fibonacci expansion in two dimensions. The accuracy of approximation is verified through various illustrative examples.The Chebyshev collocation method for a class of time fractional convection-diffusion equation with variable coefficientshttps://zbmath.org/1496.651852022-11-17T18:59:28.764376Z"Saw, Vijay"https://zbmath.org/authors/?q=ai:saw.vijay"Kumar, Sushil"https://zbmath.org/authors/?q=ai:kumar.sushilSummary: In this paper, an efficient and accurate computational scheme based on the Chebyshev collocation method and finite difference approximation is proposed to solve the time-fractional convection-diffusion equation (TFCDE) on a finite domain. The time fractional-order derivative \(\mu\in(0, 1]\) is considered in the Caputo sense. The finite-difference approximation is used in time direction while the Chebyshev collocation method is used in space direction to reduce the TFCDE into a system of algebraic equations. We also illustrate the error and convergence analysis of the proposed scheme. The proposed method is very convenient for solving such problems since the initial and boundary conditions are automatically taken into account. The efficiency and accuracy of the proposed algorithm are examined through some examples and comparisons with existing methods.Optimal computing units in the problem of discretizing solutions of the Klein-Gordon equation and their limit errorshttps://zbmath.org/1496.651942022-11-17T18:59:28.764376Z"Utesov, A. B."https://zbmath.org/authors/?q=ai:utesov.a-b"Bazarkhanova, A. A."https://zbmath.org/authors/?q=ai:bazarkhanova.a-aSummary: A specific computing unit is indicated that implements the exact order of the error that occurs when discretizing the solution of the Klein-Gordon equation by computing units constructed from exact numerical information about the initial conditions belonging to multidimensional 1-periodic Nikol'skii classes. The limit error of the specified optimal computing unit is found.Optimal strong convergence of finite element methods for one-dimensional stochastic elliptic equations with fractional noisehttps://zbmath.org/1496.652082022-11-17T18:59:28.764376Z"Cao, Wanrong"https://zbmath.org/authors/?q=ai:cao.wanrong"Hao, Zhaopeng"https://zbmath.org/authors/?q=ai:hao.zhaopeng"Zhang, Zhongqiang"https://zbmath.org/authors/?q=ai:zhang.zhongqiangSummary: We investigate the strong convergence order of piecewise linear finite element methods for a class of one-dimensional semilinear stochastic elliptic equations with additive fractional white noise. For the Hurst index \(H\in (0,1)\), we approximate the fractional Brownian motion by two spectral expansions. We show that the resulting schemes are of order \(H+1\) in the mean-square sense if the element size \(h\) is taken proportionally to the truncation parameters in the spectral approximations. Numerical results confirm our theoretical prediction.Two-dimensional Laplace transform inversion using bivariate homogeneous two-point Padé approximantshttps://zbmath.org/1496.652362022-11-17T18:59:28.764376Z"Chakir, Y."https://zbmath.org/authors/?q=ai:chakir.y"Abouir, J."https://zbmath.org/authors/?q=ai:abouir.jilali"Aounil, I."https://zbmath.org/authors/?q=ai:aounil.i"Benouahmane, B."https://zbmath.org/authors/?q=ai:benouahmane.brahimSummary: The purpose of this paper is to suggest a new numerical method for finding the inverse of the two-dimensional Laplace transform. This approach is based on bivariate homogeneous two-point Padé approximants constructed using the coefficients of the series expansions of the inverse function for small and large values. The proposed technique is verified in some numerical examples.Sparse polynomial interpolation with Bernstein polynomialshttps://zbmath.org/1496.683812022-11-17T18:59:28.764376Z"İmamoğlu, Erdal"https://zbmath.org/authors/?q=ai:imamoglu.erdalSummary: We present an algorithm for interpolating an unknown univariate polynomial \(f\) that has a \(t\) sparse representation (\(t\ll\deg(f)\)) using Bernstein polynomials as term basis from \(2t\) evaluations. Our method is based on manipulating given black box polynomial for \(f\) so that we can make use of Prony's algorithm.Constructing the bulk at the critical point of three-dimensional large \(N\) vector theorieshttps://zbmath.org/1496.810862022-11-17T18:59:28.764376Z"Johnson, Celeste"https://zbmath.org/authors/?q=ai:johnson.celeste"Mulokwe, Mbavhalelo"https://zbmath.org/authors/?q=ai:mulokwe.mbavhalelo"Rodrigues, João P."https://zbmath.org/authors/?q=ai:rodrigues.joao-pSummary: In the context of the \(AdS_4/CFT_3\) correspondence between higher spin fields and vector theories, we use the constructive bilocal fields based approach to this correspondence, to demonstrate, at the \textit{IR} critical point of the interacting vector theory and directly in the bulk, the removal of the \(\Delta = 1\) (\(s = 0\)) state from the higher spins field spectrum, and to exhibit simple Klein-Gordon higher spin Hamiltonians. The bulk variables and higher spin fields are obtained in a simple manner from boundary bilocals, by the change of variables previously derived for the \textit{UV} critical point (in momentum space), together with a field redefinition.Quark condensate and chiral symmetry restoration in neutron starshttps://zbmath.org/1496.850012022-11-17T18:59:28.764376Z"Jin, Hao-Miao"https://zbmath.org/authors/?q=ai:jin.hao-miao"Xia, Cheng-Jun"https://zbmath.org/authors/?q=ai:xia.cheng-jun"Sun, Ting-Ting"https://zbmath.org/authors/?q=ai:sun.tingting"Peng, Guang-Xiong"https://zbmath.org/authors/?q=ai:peng.guang-xiongSummary: Based on an equivparticle model, we investigate the in-medium quark condensate in neutron stars. Carrying out a Taylor expansion of the nuclear binding energy to the order of \(\rho^3\), we obtain a series of EOSs for neutron star matter, which are confronted with the latest nuclear and astrophysical constraints. The in-medium quark condensate is then extracted from the constrained properties of neutron star matter, which decreases non-linearly with density. However, the chiral symmetry is only partially restored with non-vanishing quark condensates, which may vanish at a density that is out of reach for neutron stars.Image reconstruction for positron emission tomography based on Chebyshev polynomialshttps://zbmath.org/1496.920392022-11-17T18:59:28.764376Z"Fragoyiannis, George"https://zbmath.org/authors/?q=ai:fragoyiannis.george"Papargiri, Athena"https://zbmath.org/authors/?q=ai:papargiri.athena"Kalantonis, Vassilis"https://zbmath.org/authors/?q=ai:kalantonis.vassilis-s"Doschoris, Michael"https://zbmath.org/authors/?q=ai:doschoris.michael"Vafeas, Panayiotis"https://zbmath.org/authors/?q=ai:vafeas.panayiotisSummary: The study of the functional characteristics of the brain plays a crucial role in modern medical imaging. An important and effective nuclear medicine technique is positron emission tomography (PET), whose utility is based upon the noninvasive measure of the in vivo distribution of imaging agents, which are labeled with positron-emitting radionuclides. The main mathematical problem of PET involves the inverse Radon transform, leading to the development of several methods toward this direction. Herein, we present an improved formulation based on Chebyshev polynomials, according to which a novel numerical algorithm is employed in order to interpolate exact simulated values of the Randon transform via an analytical Shepp-Logan phantom representation. This approach appears to be efficient in calculating the Hilbert transform and its derivative, being incorporated within the final analytical formulae. The numerical tests are validated by comparing the presented methodology to the well-known spline reconstruction technique.
For the entire collection see [Zbl 1485.65002].On the application of Lehmer means in signal and image processinghttps://zbmath.org/1496.940062022-11-17T18:59:28.764376Z"Amat, Sergio"https://zbmath.org/authors/?q=ai:amat.sergio-p"Magreñán, Ángel A."https://zbmath.org/authors/?q=ai:magrenan.angel-alberto"Ruiz, Juan"https://zbmath.org/authors/?q=ai:ruiz.juan-carlos|ruiz.juan-p|ruiz.juan-miguel|ruiz.juan-f"Trillo, Juan C."https://zbmath.org/authors/?q=ai:trillo.juan-carlos"Yáñez, Dionisio F."https://zbmath.org/authors/?q=ai:yanez.dionisio-fSummary: This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing.Exact reconstruction of sparse non-harmonic signals from their Fourier coefficientshttps://zbmath.org/1496.940192022-11-17T18:59:28.764376Z"Petz, Markus"https://zbmath.org/authors/?q=ai:petz.markus"Plonka, Gerlind"https://zbmath.org/authors/?q=ai:plonka.gerlind"Derevianko, Nadiia"https://zbmath.org/authors/?q=ai:derevianko.nadiiaIn this paper, the authors propose a new method to reconstruct real non-harmonic Fourier sums, i.e. real signals which can be represented as sparse exponential sums of the form
\[
f(t)=\sum_{j=1}^{K}\gamma_{j}\cos(2\pi a_j t + b_j),
\]
from their Fourier coefficients. They assume that \(K\in\mathbb{N}\), \(\gamma_j\in (0,\infty)\), \((a_j, b_j) \in (0, \infty)\times[0, 2\pi )\), and that the frequency parameters \(a_j\) are pairwise distinct.
Their approach is based on two steps. The first one consists of reconstructing the non-\(P\)-periodic part of \(f\), employing a modification of the recently proposed AAA algorithm [\textit{Y. Nakatsukasa} et al., SIAM J. Sci. Comput. 40, No. 3, A1494--A1522 (2018; Zbl 1390.41015)]; the second step concerns the determination of possible \(P\)-periodic terms of \(f\). In particular, they prove that their method allows to uniquely determine \(f\) from at most \(2K+2\) of its Fourier coefficients.
Finally, the authors present two numerical experiments, which show that the considered reconstruction scheme provides very good reconstruction results even for small frequency gaps if \(P\) is chosen suitably. These results are compared with the ones obtained with a stabilized variant of Prony's method.
Reviewer: Mariarosaria Natale (Firenze)