Recent zbMATH articles in MSC 41https://zbmath.org/atom/cc/412024-09-27T17:47:02.548271ZWerkzeugStirling's approximation and a hidden link between two of Ramanujan's approximationshttps://zbmath.org/1541.110252024-09-27T17:47:02.548271Z"O'Sullivan, Cormac"https://zbmath.org/authors/?q=ai:osullivan.cormacSummary: A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to Ramanujan's approximation coefficients. A more analytic second proof of the main result is also included in an appendix.Generalized Taylor's formula for power fractional derivativeshttps://zbmath.org/1541.260152024-09-27T17:47:02.548271Z"Zitane, Hanaa"https://zbmath.org/authors/?q=ai:zitane.hanaa"Torres, Delfim F. M."https://zbmath.org/authors/?q=ai:torres.delfim-f-mSummary: We establish a new generalized Taylor's formula for power fractional derivatives with nonsingular and nonlocal kernels, which includes many known Taylor's formulas in the literature. Moreover, as a consequence, we obtain a general version of the classical mean value theorem. We apply our main result to approximate functions in Taylor's expansions at a given point. The explicit interpolation error is also obtained. The new results are illustrated through examples and numerical simulations.On the monotonicity of left and right Riemann sumshttps://zbmath.org/1541.260392024-09-27T17:47:02.548271Z"Bouthat, Ludovick"https://zbmath.org/authors/?q=ai:bouthat.ludovickSummary: This paper is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riemann sums. Additionally, we present a new insightful proof of a fundamental theorem related to these sums using tools from the theory of majorization. Lastly, we delve deeper into a question posed by Borwein, almost resolving it completely.Dubious identities: a visit to the Borwein zoohttps://zbmath.org/1541.330022024-09-27T17:47:02.548271Z"Bradshaw, Zachary P."https://zbmath.org/authors/?q=ai:bradshaw.zachary-p"Vignat, Christophe"https://zbmath.org/authors/?q=ai:vignat.christopheSummary: We contribute to the zoo of dubious identities established by \textit{J. M. Borwein} and \textit{P. B. Borwein} in their 1992 paper, ``Strange Series and High Precision Fraud'' [Am. Math. Mon. 99, No. 7, 622--640 (1992; Zbl 0762.40003)] with five new entries, each of a different variety than the last. Some of these identities are again a high precision fraud and picking out the true from the bogus can be a challenging task with many unexpected twists along the way.
This work is dedicated to the Borwein family, mathematicians extraordinaire with a propensity for the implausible.Uniform asymptotic expansions for Lommel, Anger-Weber, and Struve functionshttps://zbmath.org/1541.330072024-09-27T17:47:02.548271Z"Dunster, T. M."https://zbmath.org/authors/?q=ai:dunster.t-mSummary: Using a differential equation approach, asymptotic expansions are rigorously obtained for Lommel, Weber, Anger-Weber, and Struve functions, as well as Neumann polynomials, each of which is a solution of an inhomogeneous Bessel equation. The approximations involve Airy and Scorer functions, and are uniformly valid for large real order \(\nu\) and unbounded complex argument \(z\). An interesting complication is the identification of the Lommel functions with the new asymptotic solutions, and in order to do so, it is necessary to consider certain sectors of the complex plane, as well as introduce new forms of Lommel and Struve functions.
{{\copyright} 2021 Wiley Periodicals LLC}On the uniqueness of an orthogonality property of the Legendre polynomialshttps://zbmath.org/1541.330082024-09-27T17:47:02.548271Z"Bos, Len"https://zbmath.org/authors/?q=ai:bos.len-p"Ware, Antony F."https://zbmath.org/authors/?q=ai:ware.antony-fSummary: Recently \textit{L. Bos} et al. [Constr. Approx. 45, No. 1, 65--81 (2017; Zbl 1362.33010)] gave a remarkable orthogonality property of the classical Legendre polynomials on the real interval \([-1, 1]\): polynomials up to degree \(n\) from this family are mutually orthogonal under the arcsine measure weighted by the degree-\(n\) normalized Christoffel function. We show that the Legendre polynomials are (essentially) the only orthogonal polynomials with this property.Asymptotics of the determinant of the modified Bessel functions and the second Painlevé equationhttps://zbmath.org/1541.330162024-09-27T17:47:02.548271Z"Chen, Yu"https://zbmath.org/authors/?q=ai:chen.yu.22|chen.yu.15|chen.yu.16|chen.yu.3|chen.yu.6|chen.yu.10|chen.yu.2|chen.yu.1|chen.yu.12|chen.yuqun|chen.yu.4|chen.yu.8"Xu, Shuai-Xia"https://zbmath.org/authors/?q=ai:xu.shuaixia"Zhao, Yu-Qiu"https://zbmath.org/authors/?q=ai:zhao.yuqiuSummary: In the paper, we consider the extended Gross-Witten-Wadia unitary matrix model by introducing a logarithmic term in the potential. The partition function of the model can be expressed equivalently in terms of the Toeplitz determinant with the \((i,j)\)-entry being the modified Bessel functions of order \(i-j-\nu, \nu\in\mathbb{C}\). When the degree \(n\) is finite, we show that the Toeplitz determinant is described by the isomonodromy \(\tau\)-function of the Painlevé III equation. As a double scaling limit, we establish an asymptotic approximation of the logarithmic derivative of the Toeplitz determinant, expressed in terms of the Hastings-McLeod solution of the inhomogeneous Painlevé II equation with parameter \(\nu+\frac{1}{2}\). The asymptotics of the leading coefficient and recurrence coefficient of the associated orthogonal polynomials are also derived. We obtain the results by applying the Deift-Zhou nonlinear steepest descent method to the Riemann-Hilbert problem for orthogonal polynomials on the Hankel loop. The main concern here is the construction of a local parametrix at the critical point \(z=-1\), where the \(\psi\)-function of the Jimbo-Miwa Lax pair for the inhomogeneous Painlevé II equation is involved.Geometric representation of the weighted harmonic mean of \(n\) positive values and potential applications.https://zbmath.org/1541.410012024-09-27T17:47:02.548271Z"Amat, S."https://zbmath.org/authors/?q=ai:amat-plata.sergio"Ortiz, P."https://zbmath.org/authors/?q=ai:ortiz.pedro"Ruiz, J."https://zbmath.org/authors/?q=ai:ruiz.j"Trillo, J. C."https://zbmath.org/authors/?q=ai:trillo.juan-carlos"Yáñez, D. F."https://zbmath.org/authors/?q=ai:yanez.dionisio-fAuthors' abstract: This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of \(n\) positive real numbers. Together with this interpretation, we prove some relevant properties that will allow us to define numerical approximation methods in several dimensions adapted to discontinuities.
Reviewer: Antonio López-Carmona (Granada)Approximation of piecewise smooth functions by nonlinear bivariate \(C^2\) quartic spline quasi-interpolants on criss-cross triangulationshttps://zbmath.org/1541.410022024-09-27T17:47:02.548271Z"Aràndiga, Francesc"https://zbmath.org/authors/?q=ai:arandiga.francesc"Remogna, Sara"https://zbmath.org/authors/?q=ai:remogna.saraThe authors focus on the space of \(C^2\) quartic splines on uniform criss-cross triangulations and propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high order accuracy in smooth regions.
Reviewer: Zhihua Zhang (Jinan)Rate of convergence of multinode Shepard operatorshttps://zbmath.org/1541.410032024-09-27T17:47:02.548271Z"Dell'Accio, Francesco"https://zbmath.org/authors/?q=ai:dellaccio.francesco"Di Tommaso, Filomena"https://zbmath.org/authors/?q=ai:di-tommaso.filomenaSummary: The triangular Shepard method, introduced by \textit{F. Little} in [``Convex combination surfaces'', in: Surfaces in computer aided geometric design. 99--108 (1983)], is a convex combination of triangular basis functions with linear polynomials, based on the vertices of the triangles, that locally interpolate the given data at the vertices. The method has linear precision and reaches quadratic approximation order [\textit{F. Dell'Accio} et al., IMA J. Numer. Anal. 36, No. 1, 359--379 (2016; Zbl 1335.65016)]. As specified by Little, the triangular Shepard method can be generalized to higher dimensions and to sets of more than three points. In this paper we introduce the multinode Shepard method as a generalization of the triangular Shepard method in the case of scattered points in \(\mathbb{R}^s\), \(s \in\mathbb{N}\), and we study the remainder term and its asymptotic behavior.Calculation of the limit of a special sequence of trigonometric functionshttps://zbmath.org/1541.410042024-09-27T17:47:02.548271Z"Alferova, E. D."https://zbmath.org/authors/?q=ai:alferova.e-d"Sherstyukov, V. B."https://zbmath.org/authors/?q=ai:sherstyukov.vladimir-borisovichLet
\[
r_n(p) = \left(\sup_{x \in \mathbb{R}} \prod_{k=0}^{n-1} |\sin(p^kx)|\right)^{1/n}.
\]
It is shown that the limit \(\lim_{n\to \infty} r_n(p)=r(p)\) does exist for all \(p \in \mathbb{R}\). The rate of the convergence is also estimated for \(p=2\).
Reviewer: Ferenc Weisz (Budapest)Approximating continuous functions by polynomialshttps://zbmath.org/1541.410052024-09-27T17:47:02.548271Z"Bharali, Gautam"https://zbmath.org/authors/?q=ai:bharali.gautamFor the entire collection see [Zbl 1242.00057].Generalized Voronovskaya theorem and the convergence of power series of positive linear operatorshttps://zbmath.org/1541.410062024-09-27T17:47:02.548271Z"Garoiu, Ştefan"https://zbmath.org/authors/?q=ai:garoiu.stefan-lucian"Păltănea, Radu"https://zbmath.org/authors/?q=ai:paltanea.raduVoronovskaya's theorem provides an asymptotic error term for the Bernstein polynomials of functions that are twice differentiable. There is an extensive body of literature on Voronovskaya-type results for various operators. The aim of the present manuscript is to generalize Voronovskaya's theorem by providing an explicit form of the limit \(\lim_{n\to\infty} n^s\left(L_n - I\right)^s\), where \(s\) is a positive integer and the operators \(L_n\) belong to a broad class of positive linear operators. This result is equivalent to explicit Voronovskaya theorems for Micchelli combinations of \(L_n\). The authors also explore a class of generalized power series of operators.
Reviewer: Cătălin Badea (Villeneuve d'Ascq)Explicit algebraic solution of Zolotarev's first problem for low-degree polynomials. IIhttps://zbmath.org/1541.410072024-09-27T17:47:02.548271Z"Rack, Heinz-Joachim"https://zbmath.org/authors/?q=ai:rack.heinz-joachim"Vajda, Robert"https://zbmath.org/authors/?q=ai:vajda.robertSummary: With recourse to [\textit{H.-J. Rack} and \textit{R. Vajda}, J. Numer. Anal. Approx. Theory 48, No. 2, 175--201 (2019; Zbl 1463.41016)], we consider three algorithms for explicitly solving, by algebraic means, Zolotarev's First Problem (ZFP) of 1868 which is described e.g. in [\textit{N. I. Achieser}, Theory of approximation. Translated from the Russian 1947 edition by Charles J. Hyman. New York: Frederick Ungar Publishing Co (1956; Zbl 0072.28403); \textit{B. C. Carlson} and \textit{J. Todd}, Aequationes Math. 26, 1--33 (1983; Zbl 0535.41029); \textit{G. V. Milovanović} et al., Topics in polynomials: extremal problems, inequalities, zeros. Singapore: World Scientific (1994; Zbl 0848.26001); \textit{S. Paszkowski}, Diss. Math. 26, 176 p. (1962; Zbl 0101.28802)]. We avoid the application of elliptic functions by drawing first on three tentative forms \(Z_{n,s,\alpha,\beta} (1<\alpha <\beta)\) of the sought-for monic proper Zolotarev polynomial \(Z_{n,s} (n \geq 4,s > \tan^2 (\pi/(2n)))\). In order to compute then the compatible \(\alpha = \alpha_0\) and \(\beta = \beta_0\), so that \(Z_{n_0, s_0, \alpha_0, \beta_0} = Z_{n_0, s_0}\) will hold for a prescribed degree \(n = n_0\) and prescribed intrinsic parameter \(s = s_0\), we draw on three intertwined variants and deploy them exemplarily to the third tentative form (not considered in [\textit{H.-J. Rack} and \textit{R. Vajda}, J. Numer. Anal. Approx. Theory 48, No. 2, 175--201 (2019; Zbl 1463.41016)]). We conclude that our first tentative form constitutes, in conjunction with our third variant, a deterministic algebraic algorithm for solving ZFP, which is advantageous with respect to complexity reduction. Three related algebraic algorithms from literature for solving ZFP, [\textit{V. A. Malyshev}, St. Petersbg. Math. J. 13, No. 6, 893--938 (2002; Zbl 1051.30009); translation from Algebra Anal. 13, No. 6, 1--55 (2001); \textit{F. Peherstorfer}, Acta Math. Hung. 55, No. 3--4, 245--278 (1990; Zbl 0726.33006); \textit{K. Schiefermayr}, J. Approx. Theory 148, No. 2, 148--157 (2007; Zbl 1185.41005)], are examined, refined and exemplified. Further existing non-elliptic approaches to ZFP, including the one by means of parametrization of algebraic curves [\textit{H.-J. Rack} and \textit{R. Vajda}, Adv. Stud.: Euro-Tbil. Math. J. 14, No. 4, 37--60 (2021; Zbl 1487.41006)], are referenced and annotated. Explicit representations of \(Z_{n,s}\) in the algebraic power form (unexampled if \(n>7\)) and novel characteristics, which facilitate the algebraic construction of \(Z_{n,s}\), are provided and additionally stored, for \(n \leq 13\), in an online ZFP-repository.An extremal problem and inequalities for entire functions of exponential typehttps://zbmath.org/1541.410082024-09-27T17:47:02.548271Z"Chirre, Andrés"https://zbmath.org/authors/?q=ai:chirre.andres"Dimitrov, Dimitar K."https://zbmath.org/authors/?q=ai:dimitrov.dimitar-k"Quesada-Herrera, Emily"https://zbmath.org/authors/?q=ai:quesada-herrera.emily"Sousa, Mateus"https://zbmath.org/authors/?q=ai:sousa.mateusThe task addressed in this article is bounding integrals of functions of exponential type \(f\) by their values at the origin; those values must not be zero and the integrands nonnegative. The integrals are over the whole real axis. This makes them a so-called one-delta-function. The infimum \(\mathcal{A}\) of these upper bounds are also required. In the classical case the solution is \(\mathcal{A}=1\) and the argmin function is the square of the sinc function.
An advanced problem of this type is when the integrands are explicitly required to be radially monotone, i.e., increasing/decreasing for the arguments going from \(-\infty\) to \(0\) and from the origin to \(+\infty\), respectively. This problem is solved with \(\mathcal{A}\in(1.2750,1.27714)\).
Reviewer: Martin D. Buhmann (Gießen)Optimal polynomial meshes exist on any multivariate convex domainhttps://zbmath.org/1541.410092024-09-27T17:47:02.548271Z"Dai, Feng"https://zbmath.org/authors/?q=ai:dai.feng"Prymak, Andriy"https://zbmath.org/authors/?q=ai:prymak.andriy-vThe identification of sequences of finite sets of compact domains that allow to bound polynomials uniformly is of great interest in analysis and algebra. The task is to find sequences of ``small'' sets \(Y_n\) so that, for a compact domain \(\Omega\), we can estimate \(\|P\|_\Omega=\|P\|_{\infty,\Omega}\) from above as
\[
O(\|P\|_{Y_n})=O(\max_{Y_n}|P(\cdot)|)
\]
for all \(P\).
The \(P\)s are polynomials of some fixed total degree (call the degree \(n\)). So the sets \(Y_n=\{x_1,x_2,\ldots,x_N\}\), \(N=N(n,d)\), depend on the degree of the polynomials and of course on the dimension.
The importance lies in finding least bounds on the size of the sets \(Y_n\). In this excellent contribution an upper bound of \(O(n^d)\) is given for the cardinality of the sets, where \(d\) is the space dimension. The \(O(\cdot)\) in \(\|P\|_\Omega\) bounded by \(O(\|P\|_{Y_n})\) is even specified as 2. This settles a conjecture due to Andras Kroo.
Reviewer: Martin D. Buhmann (Gießen)Bernstein inequality for the Riesz derivative of order \(0<\alpha<1\) of entire functions of exponential type in the uniform normhttps://zbmath.org/1541.410102024-09-27T17:47:02.548271Z"Leont'eva, A. O."https://zbmath.org/authors/?q=ai:leonteva.anastasiya-oLet \(B_\sigma\) be the class, introduced by S.Bernstein, of entire functions of the exponential type not exceeding \(\sigma \) that are bounded on the real axis. For a function \(f\in B_\sigma\) the Riesz derivative of order \(\alpha,\ 0<\alpha<1,\) is determined by the formula
\[
D^\alpha f(x)=-\pi^{-1}\Gamma (\alpha +1)\sin \pi\alpha/2\int_0^\infty\frac{f(x+y)-2f(x)+f(x-y)}{y^{\alpha +1}}dy.
\]
For this operator, the corresponding interpolation formula is obtained. With the help of this formula the Bernstein inequality
\[
\|D^\alpha f\|\leq B_\sigma (\alpha)\|f\|,\ f\in B_\sigma,
\]
in the uniform on the real line norm is studied. The extremal entire function for this inequality and exact value of \(B_\sigma (\alpha)\) are obtained.
Reviewer: D. K. Ugulava (Tbilisi)\(\mathcal{H}_2\) optimal rational approximation on general domainshttps://zbmath.org/1541.410112024-09-27T17:47:02.548271Z"Borghi, Alessandro"https://zbmath.org/authors/?q=ai:borghi.alessandro"Breiten, Tobias"https://zbmath.org/authors/?q=ai:breiten.tobiasIf \(h(z)\) is the degree \(n\) rational transfer function of a large-scale continuous single-input single-output (SISO) linear time-invariant (LTI) system then the best (in \(L^2\)-sense) stable degree \(r\) reduced order system \(\hat{h}\), can be obtained by minimizing \(\|h-\hat{h}\|_{H^2(\mathbb{C}_+)}\) in the Hardy space of the the open right half plane \(\mathbb{C}_+\), hence forcing all its poles to be in the closed left-half plane. This can be done using a Petrov-Galerkin method, finding \(\hat{h}\) by projecting the system on a lower dimensional space which can be obtained using an iterated rational Krylov algorithm (IRKA). In this method estimates $\{\hat{\lambda}_j\}_{j=1}^r$ of $\hat{h}$'s poles are iteratively improved. For a local optimal solution the following first order optimality conditions hold \((h-\hat{h})(-\hat\lambda_j)=(h'-\hat{h}')(-\hat\lambda_j)=0\), \(j=1,\ldots,r\) (the prime means derivative). A similar method exists for discrete-time systems where the open left half plane is replaced by the open unit disk. In this paper this idea is generalized to a closed region \(\bar{\mathbb{A}}\) where \(\hat{h}\) is allowed to have poles. Therefore a conformal map has to be defined that maps the complement \(\bar{\mathbb{A}}^c\) to \(\mathbb{C}_+\) to solve the problem in \(H^2(\mathbb{C}_+)\). This is elaborated when \(\mathbb{A}\) is the upper half plane or an ellipse. Several numerical examples illustrate the idea.
Reviewer: Adhemar Bultheel (Leuven)Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systemshttps://zbmath.org/1541.410122024-09-27T17:47:02.548271Z"Gosea, Ion Victor"https://zbmath.org/authors/?q=ai:gosea.ion-victor"Gugercin, Serkan"https://zbmath.org/authors/?q=ai:gugercin.serkan"Werner, Steffen W. R."https://zbmath.org/authors/?q=ai:werner.steffen-w-r\S1. Introduction (2 pages)
The authors focus their attention on systems described by the following type of differential equations using second order time derivatives of the form (1) \(\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{D}\dot{\mathbf{x}}(t)+\mathbf{K}\mathbf{x}(t)=\mathbf{b}u(t)\) \(y(t)=\mathbf{c}T\mathbf{x}(t)\) where \(\mathbf{M, D, K}\in\mathbb{R}^{n\times n}\) and \(\mathbf{b}, c\in\mathbb{R}^n\).
In the corresponding frequency domain (Laplace domain) the input-to-output behavior is given by the transfer function (2) \(H(s)=\mathbf{c}\Gamma(s^2\mathbf{M}+s\mathbf{D}+\mathbf{k})^{-1}\mathbf{b}\), a second-degree-2n ratonal function in \(s\) (\(n\) is the state space dimension of (1)).
The authors then discuss the approaches used in recent years, referring to 27 references (out of the list of 52). In this paper they develop new barycentric forms associated with the transfer function (2).
{\S2. Mathematical preliminaries and first-order systems} (\(3\frac{1}{2}\) pages)
{\S3. Structured barycentric forms} (10 pages)
Under two sets of assumptions concerning the modelmatrix and the interpolation points, the authors give their main results in the Theorems 1 and 2.
{\S4. Computational aspects and procedures} (7 pages)
Here three different algorithms are defined.
{\S5. Numerical experiments} (5 pages)
Contains five sets of two graphs each (frequency \(\omega\) versus magnitude \(|H(\omega)|\) and versus relative error).
{\S6. Conclusion} (1 page)
{References} (52 items)
Reviewer: Marcel G. de Bruin (Heemstede)On the approximation of the \(| \sin x|^s\) function by rational trigonometric operators of the Fejér typehttps://zbmath.org/1541.410132024-09-27T17:47:02.548271Z"Kazloŭskaya, Natallya Yur'eŭna"https://zbmath.org/authors/?q=ai:kazlouskaya.natallya-yureuna"Roŭba, Yaŭgen Alyakseevich"https://zbmath.org/authors/?q=ai:rovba.evgenii-alekseevichLet \( \alpha_k\in [0,1)\), \(\alpha_{n+k}=-\alpha_k\), \(k=1,2,\ldots, n\), and let
\[
\lambda_{2n}(u)=\frac{1}{2}+\frac{1}{2} \sum_{k=1}^{2n} \frac{1-|\alpha_k|^2}{1-2|\alpha_k| \cos (u-\mathrm{arg}\, \alpha_k)+|\alpha_k|^2}.
\]
For an arbitrary continuous \(2\pi\)-periodic function \(f\), we set
\[
\Phi_{2n}(x,f)=\frac{\int_{-\pi}^{\pi}f(t)D_{2n}^2(t,x)dt}{\int_{-\pi}^{\pi}D_{2n}^2(t,x)dt},\quad
x\in \mathbb{R},
\]
where
\[
D_{2n}(t,x)=\frac{\sin \lambda_{2n}(t,x)}{\sin\frac{t-x}{2}},\quad \lambda_{2n}(t,x)=\int_{x}^{t}\lambda_{2n}(u)du.
\]
The paper studies the asymptotic behavior of the value
\[
\varepsilon_{2n}(x,\alpha)=|\sin x|^s - \Phi_{2n}(x,|\sin x|^s),\quad x\in \mathbb{R},
\]
where \(s\in (0;2)\), \(\alpha=(\alpha_1,\ldots, \alpha_{2n})\). The authors obtain an integral representation for \(\varepsilon_{2n}(x,\alpha)\) on the basis of which they come to the following result.
{Theorem.} Let \(x\in (-\pi,0)\cup (0,\pi)\), \( \alpha_k\in [0,1)\), \(k=1,2,\ldots,\) and assume that
\[
\sum_{k=1}^{\infty} (1-|\alpha_k|)=+\infty.
\]
Then as \(n\rightarrow +\infty\),
\[
\varepsilon_{2n}(x,\alpha)=-\frac{2^{2-s} \sin\frac{\pi s}{2} }{\pi \lambda_{2n}(x)} \left(\int_{0}^{1}t^{1-s}(1-t^2)^s \frac{(t^4+1)\cos 2x -2t^2}{(t^4-2t^2 \cos 2x+1)^2} dt +o(1)\right).
\]
Next, the case when \(\alpha_1=\ldots= \alpha_{n}=\alpha\) is studied in more detail.
Reviewer: Vitaliy Volchkov (Donetsk)Asymptotics of Hermitian approximation of exponentshttps://zbmath.org/1541.410142024-09-27T17:47:02.548271Z"Starovoitov, A. P."https://zbmath.org/authors/?q=ai:starovoitov.aleksandr-pavlovichSummary: The paper deals with asymptotic properties of Hermite integrals. In particular, the asymptotics of Hermite-Padé diagonal approximations for the system of exponents are determined. Similar results are proved for the system of confluent hypergeometric functions.On extreme property of quadratic Hermite-Padé approximantshttps://zbmath.org/1541.410152024-09-27T17:47:02.548271Z"Starovoitov, A. P."https://zbmath.org/authors/?q=ai:starovoitov.aleksandr-pavlovichSummary: The extreme properties of diagonal quadratic Hermite-Padé approximants of type I for exponential system \(\left\{e^{\lambda_jz}\right\}^2_{j=0}\) with arbitrary real \(\lambda_0,\lambda_1,\lambda_2\) are studied. The proved theorems complement the known results of P. Borwein and F. Wielonsky.Approximation results by multivariate Kantorovich-type neural network sampling operators in Lebesgue spaces with variable exponentshttps://zbmath.org/1541.410162024-09-27T17:47:02.548271Z"Aharrouch, Benali"https://zbmath.org/authors/?q=ai:aharrouch.benaliThe problem of the convergence of a family of neural network operators of the type Kantorovich with sigmoidal activation functions in Lebesgue spaces \(L^{p(\cdot)}(\mathfrak{R})\) with variable exponents is studied. It is assumed that \(\mathfrak{R}:=[c_1,d_1]\times \cdots \times [c_m,d_m] \subset \mathbb{R}^m\) and \(1\leq p^-\leq p(x)\leq p^+ <+\infty,\ x\in \mathfrak{R}\). Also, the pointwise and uniform convergence for functions belonging to suitable spaces are proved. Note that the \(L^{p(\cdot)}\) convergence is established by using the density of the set of functions in such spaces. Some examples of sigmoidal activation functions for which the present theory can be applied are presented. The convergence of the Kantorovich approximations of \(L^{p(\cdot)}\) functions are illustrated by graphical representation.
Reviewer: D. K. Ugulava (Tbilisi)A maximal oscillatory operator on compact manifoldshttps://zbmath.org/1541.410172024-09-27T17:47:02.548271Z"Liu, Ziyao"https://zbmath.org/authors/?q=ai:liu.ziyao"Chen, Jiecheng"https://zbmath.org/authors/?q=ai:chen.jiecheng"Fan, Dashan"https://zbmath.org/authors/?q=ai:fan.dashanOscillatory integral operators \(T_{\alpha,\beta}\) are investigated on compact manifolds \(\mathbb{M}\). It is proved that the maximal operator \(T_{\alpha,\beta}^*\) is bounded from the Hardy space \(H^p\) to \(L^{p,\infty}\) \((0<p<1)\) together with its sharpness. As applications, they obtain inequalities and convergence results for Riesz means associated with the Schroödinger type group.
Reviewer: Ferenc Weisz (Budapest)Variable transformations in combination with wavelets and ANOVA for high-dimensional approximationhttps://zbmath.org/1541.410182024-09-27T17:47:02.548271Z"Potts, Daniel"https://zbmath.org/authors/?q=ai:potts.daniel"Weidensager, Laura"https://zbmath.org/authors/?q=ai:weidensager.lauraLet \(\Omega \in \{{\mathbb T}^d,\,{\mathbb R}^d,\, [0,\,1]^d\}\) or tensor products of these domains. The authors consider the approximation of a high-dimensional function \(f:\,\Omega \to \mathbb C\) from discrete sample points, which are distributed to an arbitrary density. They show that it is possible to transform the good approximation results for periodic functions on the torus \({\mathbb T}^d\) (see [\textit{L. Lippert} et al., Numer. Math. 154, No. 1--2, 155--207 (2023; Zbl 1528.41030)]) to the domain \(\Omega\). This method combines the least squares approximation on \({\mathbb T}^d\) and the truncation of the ANOVA (analysis of variance) decomposition with a variable transformation and a density estimation. The error decay rates and fast algorithms are transferred from the torus \({\mathbb T}^d\) to the domain \(\Omega\). A new extension method, which benefits from the Chui-Wang wavelets, allows the approximation of non-periodic functions too. Numerical experiments illustrate the performance of these procedures.
Reviewer: Manfred Tasche (Rostock)Approximation by modified generalized sampling serieshttps://zbmath.org/1541.410192024-09-27T17:47:02.548271Z"Turgay, Metin"https://zbmath.org/authors/?q=ai:turgay.metin"Acar, Tuncer"https://zbmath.org/authors/?q=ai:acar.tuncerUsing Bernstein polynomials is one of the important tools to prove the well-known Weierstrass approximation theorem for the space of continuous functions on \([0,1] \) or more generally on \( [a, b] \subset {\mathbb{R}} \) [\textit{S. Bernstein}, Charkow Ges. (2) 13, 1--2 (1912; JFM 43.0301.03)]. In [Acta Math. Hung. 99, No. 3, 203--208 (2003; Zbl 1027.41028)], \textit{J. P. King} constructed a generalization of the classical Bernstein operators using a sequence of continuous functions defined on \([0,1] \) to obtain a better approximation.
In the paper under review, the authors construct a new form of generalized sampling operators by considering a \( \rho \) function, which satisfies some required conditions. They also give examples of graphical representations and numerical tables to compare the modified sampling operators and the classical sampling operators using the central B-spline kernel, although the results can also be obtained by taking the other kernels which satisfy the assumptions of Theorem 1. They note that the examples show that newly constructed operators are better in approach than the old ones in some cases.
Reviewer: Hüseyin Çakallı (İstanbul)Approximation properties of exponential sampling series in logarithmic weighted spaceshttps://zbmath.org/1541.410202024-09-27T17:47:02.548271Z"Acar, Tuncer"https://zbmath.org/authors/?q=ai:acar.tuncer"Kursun, Sadettin"https://zbmath.org/authors/?q=ai:kursun.sadettin"Acar, Özlem"https://zbmath.org/authors/?q=ai:acar.ozlemThe authors study the convergence properties of a family of exponential sampling series in the space of continuous functions with respect to a logarithmic weight function. The pointwise and uniform convergence of exponential sampling series in weighted spaces are established, and the rates of convergence via a suitable modulus of continuity in logarithmic weighted spaces are given. A quantitative representation of the pointwise asymptotic behavior of these series using Mellin-Taylor's expansion is also obtained.
Reviewer: Zoltán Finta (Cluj-Napoca)The Peano-Sard theorem for Caputo fractional derivatives and applicationshttps://zbmath.org/1541.410212024-09-27T17:47:02.548271Z"Fernandez, Arran"https://zbmath.org/authors/?q=ai:fernandez.arran"Buranay, Suzan Cival"https://zbmath.org/authors/?q=ai:buranay.suzan-civalSummary: The classical Peano-Sard theorem is a very useful result in approximation theory, bounding the errors of approximations that are exact on sets of polynomials. A fractional version was developed by \textit{K. Diethelm} [Numer. Funct. Anal. Optim. 18, No. 7--8, 745--757 (1997; Zbl 0892.41018)] for fractional derivatives of Riemann-Liouville type, which we here extend to fractional derivatives of Caputo type. We indicate some applications to quadrature and interpolation formulae. These results will be useful in the approximate solution of fractional differential equations involving Caputo-type operators, which are often said to be more natural for applications.Encoding of data sets and algorithmshttps://zbmath.org/1541.410222024-09-27T17:47:02.548271Z"Doctor, Katarina"https://zbmath.org/authors/?q=ai:doctor.katarina"Mao, Tong"https://zbmath.org/authors/?q=ai:mao.tong"Mhaskar, Hrushikesh"https://zbmath.org/authors/?q=ai:mhaskar.hrushikesh-nThe insurance of the quality of the output of a machine learning algorithm as well as its reliability in comparison to the complexity of the algorithm used constitutes a key element in many applications. In this contribution, the authors present a mathematically rigorous theory in order to take a decision about what models as algorithms applied on data sets are close to each other in terms of certain metrics, such as performance and the complexity level of the algorithm. The mathematical control of the decision is given in terms of metric entropy and capacity on certain functional spaces.
Given a normed linear space \(X\) and \(K,\) a compact subset, for any \(\epsilon>0\) let \(\mathcal{N}_{\epsilon}(K)\) be the minimal value of \(n\) such that there exists an \(\epsilon\)-net of \(K\) consisting of \(n\) points. The entropy of \(K\) is defined as \(H_{\epsilon}(K)= log \mathcal{N}_{\epsilon}(K).\) On the other hand, let \(\mathcal{M}_{\epsilon}(K)\) be the maximum value of \(m\) for which there exist \(m\) \(\epsilon\)-separable points for \(K\). The capacity of \(K\) is defined as \(C_{\epsilon}(K)= log \mathcal{M}_{\epsilon}(K).\)
Analytic functions on an ellipsoid in \(\mathbb{C}^{q}\) as well as entire functions of exponential type defined in \(\mathbb{C}^{Q}\) are introduced. In such a way, compact spaces of analytic and entire functions are defined. Estimates of the metric entropy of a compact subset of functions of infinitely many variables that arise in the definition of these spaces constitute the core of the contributions of the paper under review.
The metric entropy and capacity related to a finite number of balls of a given radius to cover a compact set are the basic tools in order to get lower and upper bounds for such estimates. On the other hand, the probability measures analyzed therein have densities which are analytic while the functionals are entire functions of exponential type defined on an infinite-dimensional sequence space. These functionals can be viewed as acting on the sequence of Fourier coefficients of the input function with respect to multivariate Chebyshev polynomials.
Finally, computational issues are discussed in the framework of the generation of analytic and bandlimited functions as well as the generation of \(\epsilon\)-nets on ellipsoids, see Kernel-based analysis of massive data [\textit{H. N. Mhaskar}, Front. Appl. Math. Stat. 6, Article ID 30, 18 p. (2020; \url{doi:10.3389/fams.2020.00030})].
Reviewer: Francisco Marcellán (Leganes)Kolmogorov widths of an intersection of a finite family of Sobolev classeshttps://zbmath.org/1541.410232024-09-27T17:47:02.548271Z"Vasil'eva, Anastasia A."https://zbmath.org/authors/?q=ai:vasileva.anastasia-andreevnaGiven a John domain \(\Omega\subset\mathbb{R}^d\), the author obtain order estimates for the Kolmogorov widths \(d_n(M; L_q(\Omega))\), where \( M = \bigcap_{j=1}^sW_{p_j}^{r_j}(\Omega); \) that is, \(M\) is the intersection of a finite family of Sobolev classes on \(\Omega\). In addition, a generalization of the theorem from [\textit{E. M. Galeev}, Math. USSR, Izv. 36, No. 2, 435--448 (1991; Zbl 0728.42002); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 2, 418--430 (1990)] on widths of an intersection of Sobolev classes on the one-dimensional torus is proved.
Reviewer: Yuri A. Farkov (Moskva)Infinite-dimensional integration and \(L^{2}\)-approximation on Hermite spaceshttps://zbmath.org/1541.410242024-09-27T17:47:02.548271Z"Gnewuch, M."https://zbmath.org/authors/?q=ai:gnewuch.michael"Hinrichs, A."https://zbmath.org/authors/?q=ai:hinrichs.aicke"Ritter, K."https://zbmath.org/authors/?q=ai:ritter.klaus.1"Rüßmann, R."https://zbmath.org/authors/?q=ai:russmann.rThe authors continue what was discussed in a previous paper [\textit{M. Gnewuch} et al., J. Complexity 71, Article ID 101654, 40 p. (2022; Zbl 1498.46031)]. They define a sequence of Hermite kernels \(k_j(x,y)=\sum_k \alpha_{k,j}^{-1} h_k(x)h_k(y)\) for scalar functions in \(L^2(\mu_0)\) where the coefficients \(\alpha_{k,j}\) satisfy certain properties that define the smoothness of the Hermite space \(H(k_j)\).
Taking infinite tensor products \(\mu=\bigotimes_{j\in\mathbb{N}} \mu_0\), and defining the Hermite space \(H(K)=\bigotimes_{j\in\mathbb{N}} H(k_j)\) with \(K(x,y)=\prod_{j\in\mathbb{N}} k_j(x_j,y_j)\) using infinite dimensional variables \(x,y\in\mathbb{R}^{\mathbb{N}}\), the functions in \(H(K)\) have maximal domain \(\mathfrak{X}=\{x\in\mathbb{R}^{\mathbb{N}}:\sum_{j\in\mathbb{N}} k_j(x_j,x_j)<\infty\}\subsetneq\mathbb{R}^{\mathbb{N}}\).
It is in this setting of functions with infinitely many variables that the authors derive upper and lower bounds for the rate of decay of the best worst case integration and best worst case \(L^2(\mu)\) approximation error when only \(n\) function evaluations are allowed in the method. These bounds will depend on the asymptotic behavior of the \(\alpha_{k,j}\). Exact rates can be obtained for \(\alpha\)'s with polynomial or (sub)exponetial growth.
This is a generalization of the results for \(\mathbb{R}^d\) in [\textit{D. Dũng} and \textit{V. K. Nguyen}, ``Optimal numerical integration and approximation of functions on $\mathbb{R}^d$ equipped with Gaussian measure'', Preprint, \url{arXiv:2207.01155}]. The proofs depend on an embedding of \(H(K)\) and a multivariate decomposition method so that the scalar results can be transferred to the infinite dimensional case.
Reviewer: Adhemar Bultheel (Leuven)Eigenvalues of truncated unitary matrices: disk counting statisticshttps://zbmath.org/1541.410252024-09-27T17:47:02.548271Z"Ameur, Yacin"https://zbmath.org/authors/?q=ai:ameur.yacin"Charlier, Christophe"https://zbmath.org/authors/?q=ai:charlier.christophe"Moreillon, Philippe"https://zbmath.org/authors/?q=ai:moreillon.philippeLet \(n\) be a positive integer and \(\alpha > 0\). Denote by \(T\) an \(n\times n\) truncation of an \((n+\alpha)\times(n+\alpha)\) Haar distributed unitary matrix. The authors investigate the disk counting statistics of the eigenvalues of \(T\) proving that for fixed \(\alpha\), the associated moment generating function obeys an asymptotics of the form \(\exp (C_1\,n + C_2 + o(1))\) as \(n\to\infty\). The constants \(C_1\) and \(C_2\) are given in terms of the incomplete Gamma function. The proof employs the uniform asymptotics of the incomplete Beta function.
Reviewer: Peter Massopust (München)On rational approximations of Poisson integrals on the interval by Fejér sums of Fourier-Chebyshev integral operatorshttps://zbmath.org/1541.410262024-09-27T17:47:02.548271Z"Potseĭko, Pavel Gennad'Evich"https://zbmath.org/authors/?q=ai:potseiko.pavel-gennadevich"Rovba, Evgeniĭ Alekseevich"https://zbmath.org/authors/?q=ai:rovba.evgenii-alekseevichApproximations of the Fejér sums of the Fourier-Chebyshev rational integral operators with restrictions on numerical geometrically different poles are investigated. The object of research is the class of functions defined by Poisson integrals on the segment $[-1,1]$. Integral representations of approximations and upper estimates of uniform approximations are presented. In the case when the boundary function has a power singularity on the segment $[-1,1]$, upper estimates of pointwise and uniform approximations are found, and the asymptotic representation of the majorant of uniform approximations is found. As a separate problem, approximations of Poisson integrals for two geometrically different poles of the approximating rational function are considered. In this case, the optimal values of the parameters at which the highest rate of uniform approximations by the studied method is achieved are found. Asymptotic expressions of the exact upper bounds of the deviations of Fejer sums of polynomial Fourier-Chebyshev series on classes of Poisson integrals on a segment are obtained. Estimates of uniform approximations by Fejer sums of polynomial Fourier-Chebyshev series of functions given by Poisson integrals on a segment with a boundary function having a power singularity are proposed.
Reviewer: V. L. Leontiev (Sankt-Peterburg)Radius of information for two intersected centered hyperellipsoids and implications in optimal recovery from inaccurate datahttps://zbmath.org/1541.410272024-09-27T17:47:02.548271Z"Foucart, Simon"https://zbmath.org/authors/?q=ai:foucart.simon"Liao, Chunyang"https://zbmath.org/authors/?q=ai:liao.chunyangThis paper considers the problem of how to determine the optimal way to recover the linear quantities of objects belonging to a known model set and observed by a specified linear process from a worst-case point of view. It is shown that if the model set is the intersection of two hyperellipsoids centered at the origin, then there is a linear optimal recovery method.
Reviewer: Jin Liang (Shanghai)Optimal recovery and generalized Carlson inequality for weights with symmetry propertieshttps://zbmath.org/1541.410282024-09-27T17:47:02.548271Z"Osipenko, K. Yu."https://zbmath.org/authors/?q=ai:osipenko.konstantin-yuThis paper investigates the recovery of operators from noisy information in weighted \(L^q\)-spaces with homogeneous weights and presents some results. The main difference between the results of this paper and those of previous studies is that exact inequalities of Carlson type and optimal recovery methods for several weights are given in the paper.
Reviewer: Jin Liang (Shanghai)Szegő recurrence for multiple orthogonal polynomials on the unit circlehttps://zbmath.org/1541.420272024-09-27T17:47:02.548271Z"Kozhan, Rostyslav"https://zbmath.org/authors/?q=ai:kozhan.rostyslav-v"Vaktnäs, Marcus"https://zbmath.org/authors/?q=ai:vaktnas.marcusSummary: We investigate polynomials that satisfy simultaneous orthogonality conditions with respect to several measures on the unit circle. We generalize the direct and inverse Szegő recurrence relations, identify the analogues of the Verblunsky coefficients, and prove the Christoffel-Darboux formula. These results should be viewed as the direct analogue of the nearest neighbour recurrence relations from the theory of multiple orthogonal polynomials on the real line.B-spline quarklets and biorthogonal multiwaveletshttps://zbmath.org/1541.420352024-09-27T17:47:02.548271Z"Hovemann, Marc"https://zbmath.org/authors/?q=ai:hovemann.marc"Kopsch, Anne"https://zbmath.org/authors/?q=ai:kopsch.anne"Raasch, Thorsten"https://zbmath.org/authors/?q=ai:raasch.thorsten"Vogel, Dorian"https://zbmath.org/authors/?q=ai:vogel.dorianSummary: In this paper, we show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces \(V_{p,j}\) of \(L_2(\mathbb{R})\) which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters, they even satisfy all those properties. Moreover, we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore, we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from \(\mathcal{S}'(\mathbb{R})\). Finally, we also verify that quarks and quarklets can be used to define sequences of subspaces \(W_{p, j}\) of \(L_2(\mathbb{R})\) that yield non-orthogonal decompositions of \(L_2(\mathbb{R})\).Extensions of continuous linear functionals and smoothness of Banach spaceshttps://zbmath.org/1541.460032024-09-27T17:47:02.548271Z"Gayathri, P."https://zbmath.org/authors/?q=ai:gayathri.p"Shunmugaraj, P."https://zbmath.org/authors/?q=ai:shunmugaraj.p"Thota, Vamsinadh"https://zbmath.org/authors/?q=ai:thota.vamsinadhOn Phelps' extension property concerning unique Hahn-Banach extensions, we have:
\begin{itemize}
\item Every closed subspace \(Y\subset X\) has the extension property (or, \(X\) has property (U)) if and only if \(X^\ast\) is rotund (Taylor-Foguel).
\item The closed subspace \(Y\subset X\) has the extension property if and only if \(Y^\perp\) is Chebychev.
\end{itemize}
Concerning Zizler's uniform extension property we have, as demonstrated by the authors in Section~2:
\begin{itemize}
\item Every hyperplane \(Y\subset X\) has the uniform extension property if and only if \(X^\ast\) is rotund in every direction.
\item The closed subspace \(Y\subset X\) has the uniform extension property if and only if \(Y^\perp\) is uniformly strongly Chebychev for bounded sets \(M\) in \(X^\ast\) (Thm.~2.6).
\item $M$-ideals have the uniform extension property (Cor.~2.8).
\end{itemize}
Let \(Y\) be a closed subspace of \(X\). For \(f\in X^\ast\) and \(\delta\geq 0\), let
\[
E_Y(f,\delta)=\{g\in X^\ast: g|_Y=f|_Y\ \mbox{and}\ \|g\|\leq\|f|_Y\|+\delta\}.
\]
\(Y\) has the \textit{strong extension property} in \(X\) if, for every \(f\in X^\ast\) and every \(\varepsilon>0\), there exists \(\delta>0\) such that the diameter of \(E(f,\delta)\leq\varepsilon\). It is demonstrated that this new property lies properly in between the extension properties of Phelps and Zizler.
Sections 3--5 contain lots of results on how the above three extension properties are related under various conditions. As examples, they coincide for finite-dimensional \(Y\) in \(X\) and Phelps's extension property implies the strong extension property whenever \(Y\) has finite co-dimension in \(X\) or \(X^\ast\) has the Namioka-Phelps property. Moreover, it is demonstrated how they relate to smoothness properties.
Reviewer: Olav Nygaard (Kristiansand)A geometric Laplace methodhttps://zbmath.org/1541.530212024-09-27T17:47:02.548271Z"Léger, Flavien"https://zbmath.org/authors/?q=ai:leger.flavien"Vialard, François-Xavier"https://zbmath.org/authors/?q=ai:vialard.francois-xavierSummary: A classical tool for approximating integrals is the Laplace method. The first- and higher-order Laplace formulas are most often written in coordinates without any geometrical interpretation. In this article, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation to the first-order term of the Laplace method. The central tool is a metric introduced by \textit{Y.-H. Kim} and \textit{R. J. McCann} [J. Eur. Math. Soc. (JEMS) 12, No. 4, 1009--1040 (2010; Zbl 1191.49046)] in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two metrics arising naturally in the Kim-McCann framework. We give an explicitly quantified version of the Laplace formula, as well as examples of applications.A quadratic spline projection method for computing stationary densities of random mapshttps://zbmath.org/1541.650072024-09-27T17:47:02.548271Z"Alshekhi, Azzah"https://zbmath.org/authors/?q=ai:alshekhi.azzah"Ding, Jiu"https://zbmath.org/authors/?q=ai:ding.jiu"Rhee, Noah"https://zbmath.org/authors/?q=ai:rhee.noah-hThe authors present a quadratic spline projection method that computes stationary densities of Markov operators associated with given random maps with position-dependent probabilities. It is shown that this method converges in the \(L^1\)-norm. Numerical experiments illustrate the results.
Reviewer: Manfred Tasche (Rostock)Multilevel Schoenberg-Marsden variation diminishing operator and related quadratureshttps://zbmath.org/1541.650082024-09-27T17:47:02.548271Z"Fornaca, Elena"https://zbmath.org/authors/?q=ai:fornaca.elena"Lamberti, Paola"https://zbmath.org/authors/?q=ai:lamberti.paolaSummary: In this paper we propose an improvement of the classical Schoenberg-Marsden variation diminishing operator with applications to the construction of new quadrature rules that we show having better performances with respect to the already known ones based on the classical cited operator. We discuss convergence properties and error estimates. Numerical experiments are also carried out to confirm the presented theoretical results.Pointwise error estimates and local superconvergence of Jacobi expansionshttps://zbmath.org/1541.650102024-09-27T17:47:02.548271Z"Xiang, Shuhuang"https://zbmath.org/authors/?q=ai:xiang.shuhuang"Kong, Desong"https://zbmath.org/authors/?q=ai:kong.desong"Liu, Guidong"https://zbmath.org/authors/?q=ai:liu.guidong"Wang, Li-Lian"https://zbmath.org/authors/?q=ai:wang.lilianSummary: As one myth of polynomial interpolation and quadrature, \textit{L. N. Trefethen} [``Six myths of polynomial interpolation and quadrature'', Math. Today (Southend-on-Sea), 184--188 (2011)] revealed that the Chebyshev interpolation of \(|x-a|\) (with \(|a|<1\)) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about 95\% range of \([-1,1]\) except for a small neighbourhood near the singular point \(x=a\). In this paper, we rigorously show that the Jacobi expansion for a more general class of \(\Phi\)-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired \(\log n\)-factor in the pointwise error estimate for the Legendre expansion recently stated in [\textit{I. Babuška} and \textit{H. Hakula}, Comput. Methods Appl. Mech. Eng. 345, 748--773 (2019; Zbl 1440.65027)] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothnesshttps://zbmath.org/1541.650272024-09-27T17:47:02.548271Z"Griebel, Michael"https://zbmath.org/authors/?q=ai:griebel.michael"Harbrecht, Helmut"https://zbmath.org/authors/?q=ai:harbrecht.helmut"Schneider, Reinhold"https://zbmath.org/authors/?q=ai:schneider.reinholdSummary: Let \(\Omega_i\subset \mathbb{R}^{n_i}\), \(i=1, \dots, m\), be given domains. In this article, we study the low-rank approximation with respect to \(L^2(\Omega_1 \times \cdots \times \Omega_m)\) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare [\textit{M. Griebel} and \textit{H. Harbrecht}, IMA J. Numer. Anal. 34, No. 1, 28--54 (2014; Zbl 1287.65009); IMA J. Numer. Anal. 39, No. 4, 1652--1671 (2019; Zbl 1496.65020)], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection-diffusion-reaction problems with two small parametershttps://zbmath.org/1541.651182024-09-27T17:47:02.548271Z"Ansari, Khursheed J."https://zbmath.org/authors/?q=ai:ansari.khursheed-jamal"Izadi, Mohammad"https://zbmath.org/authors/?q=ai:izadi.mohammad-a|izadi.mohammad"Noeiaghdam, Samad"https://zbmath.org/authors/?q=ai:noeiaghdam.samadSummary: This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection-diffusion-reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order \(\mathcal{O}(\Delta\tau^s + M^{-\frac{1}{2}})\) for \(s = 1, 2\), where \(\Delta\tau\) is the time step and \(M\) is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.An implicit scheme for time-fractional coupled generalized Burgers' equationhttps://zbmath.org/1541.651282024-09-27T17:47:02.548271Z"Vigo-Aguiar, J."https://zbmath.org/authors/?q=ai:vigo-aguiar.jesus"Chawla, Reetika"https://zbmath.org/authors/?q=ai:chawla.reetika"Kumar, Devendra"https://zbmath.org/authors/?q=ai:kumar.devendra.1"Mazumdar, Tapas"https://zbmath.org/authors/?q=ai:mazumdar.tapas.1Summary: This article presents an efficient implicit spline-based numerical technique to solve the time-fractional generalized coupled Burgers' equation. The time-fractional derivative is considered in the Caputo sense. The time discretization of the fractional derivative is discussed using the quadrature formula. The quasilinearization process is used to linearize this non-linear problem. In this work, the formulation of the numerical scheme is broadly discussed using cubic B-spline functions. The stability of the proposed method is proved theoretically through Von-Neumann analysis. The reliability and efficiency are demonstrated by numerical experiments that validate theoretical results via tables and plots.Fast imaging of sources and scatterers in a stratified ocean waveguidehttps://zbmath.org/1541.651382024-09-27T17:47:02.548271Z"Liu, Keji"https://zbmath.org/authors/?q=ai:liu.kejiSummary: In this work, we have studied the asymptotic behavior of Green's function and the reciprocity relation of the far-field pattern in the stratified ocean waveguide. Moreover, two direct sampling methods (DSM) are proposed to determine the marine sources and scatterers from the far-field data. The direct approaches are fast, easy to implement, and computationally efficient since they involve only scalar product but no matrix inversion. In the numerical simulations, the DSM for the source is capable of identifying the sources from very few observation data, and the DSM for the scatterer can reconstruct the scatterers in different shapes, scales, types, and positions. The effectiveness and robustness of the novel methods are also demonstrated. Thus, the DSM can be viewed as simple and efficient numerical techniques for providing reliable initial approximate locations of the marine sources and scatterers for any existing more refined and advanced but computationally more demanding algorithms to recover the accurate physical profiles.Unfitted Trefftz discontinuous Galerkin methods for elliptic boundary value problemshttps://zbmath.org/1541.651532024-09-27T17:47:02.548271Z"Heimann, Fabian"https://zbmath.org/authors/?q=ai:heimann.fabian"Lehrenfeld, Christoph"https://zbmath.org/authors/?q=ai:lehrenfeld.christoph"Stocker, Paul"https://zbmath.org/authors/?q=ai:stocker.paul"von Wahl, Henry"https://zbmath.org/authors/?q=ai:von-wahl.henrySummary: We propose a new geometrically unfitted finite element method based on discontinuous Trefftz ansatz spaces. Trefftz methods allow for a reduction in the number of degrees of freedom in discontinuous Galerkin methods, thereby, the costs for solving arising linear systems significantly. This work shows that they are also an excellent way to reduce the number of degrees of freedom in an unfitted setting. We present a unified analysis of a class of geometrically unfitted discontinuous Galerkin methods with different stabilisation mechanisms to deal with small cuts between the geometry and the mesh. We cover stability and derive a-priori error bounds, including errors arising from geometry approximation for the class of discretisations for a model Poisson problem in a unified manner. The analysis covers Trefftz and full polynomial ansatz spaces, alike. Numerical examples validate the theoretical findings and demonstrate the potential of the approach.A high-order shifted boundary virtual element method for Poisson equations on 2D curved domainshttps://zbmath.org/1541.651552024-09-27T17:47:02.548271Z"Hou, Yongli"https://zbmath.org/authors/?q=ai:hou.yongli"Liu, Yi"https://zbmath.org/authors/?q=ai:liu.yi.12"Wang, Yanqiu"https://zbmath.org/authors/?q=ai:wang.yanqiuSummary: We consider a high-order virtual element method for Poisson problems with non-homogeneous Dirichlet boundary condition on 2D domains with curved boundary. The scheme is designed on unfitted polygonal meshes. It borrows the idea of the shifted boundary method proposed by \textit{A. Main} and \textit{G. Scovazzi} [J. Comput. Phys. 372, 972--995 (2018; Zbl 1415.76457)] for treating the curved boundary. We prove the stability and the optimal error estimate in energy norm for the proposed method. For the \(L^2\) norm, although suboptimal error estimate is proved theoretically, numerical results appear to be optimal. Supporting numerical results are presented.A synchronous NPA hierarchy with applicationshttps://zbmath.org/1541.810292024-09-27T17:47:02.548271Z"Russell, Travis B."https://zbmath.org/authors/?q=ai:russell.travis-bSummary: We present an adaptation of the NPA hierarchy to the setting of synchronous correlation matrices. Our adaptation improves upon the original NPA hierarchy by using smaller certificates and fewer constraints, although it can only be applied to certify synchronous correlations. We recover characterizations for the sets of synchronous quantum commuting and synchronous quantum correlations. For applications, we show that the existence of symmetric informationally complete positive operator-valued measures and maximal sets of mutually unbiased bases can be verified or invalidated with only two certificates of our adapted NPA hierarchy.A contribution to the mathematical theory of diffraction: a note on double Fourier integralshttps://zbmath.org/1541.810372024-09-27T17:47:02.548271Z"Assier, R. C."https://zbmath.org/authors/?q=ai:assier.raphael-c"Shanin, A. V."https://zbmath.org/authors/?q=ai:shanin.andrey-v"Korolkov, A. I."https://zbmath.org/authors/?q=ai:korolkov.andrey-iSummary: We consider a large class of physical fields \(u\) written as double inverse Fourier transforms of some functions \(F\) of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of \(u\). In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form far-field asymptotic component of \(u\). We conclude the article by validating this theory against full numerical computations for two specific examples.Borel summability of the \(1/N\) expansion in quartic \(\mathrm{O}(N)\)-vector modelshttps://zbmath.org/1541.810852024-09-27T17:47:02.548271Z"Ferdinand, L."https://zbmath.org/authors/?q=ai:ferdinand.leonard"Gurau, R."https://zbmath.org/authors/?q=ai:gurau.razvan-g"Perez-Sanchez, C. I."https://zbmath.org/authors/?q=ai:perez-sanchez.carlos-ignacio"Vignes-Tourneret, F."https://zbmath.org/authors/?q=ai:vignes-tourneret.fabienSummary: We consider a quartic \(\mathrm{O}(N)\)-vector model. Using the loop vertex expansion, we prove the Borel summability in \(1/N\) along the real axis of the partition function and of the connected correlations of the model. The Borel summability holds uniformly in the coupling constant, as long as the latter belongs to a cardioid like domain of the complex plane, avoiding the negative real axis.QDT -- a Matlab toolbox for the simulation of coupled quantum systems and coherent multidimensional spectroscopyhttps://zbmath.org/1541.810862024-09-27T17:47:02.548271Z"Kenneweg, Tristan"https://zbmath.org/authors/?q=ai:kenneweg.tristan"Mueller, Stefan"https://zbmath.org/authors/?q=ai:muller.stefan.10|muller.stefan.2|muller.stefan.3|muller.stefan.5|mueller.stefan|muller.stefan.8|muller.stefan.4|muller.stefan.9|muller.stefan.6|muller.stefan.7|muller.stefan-c|muller-arisona.stefan|muller.stefan.1"Brixner, Tobias"https://zbmath.org/authors/?q=ai:brixner.tobias"Pfeiffer, Walter"https://zbmath.org/authors/?q=ai:pfeiffer.walterSummary: We present QDT (``quantum dynamics toolbox''), an open-source Matlab software package that enables users to simulate coupled quantum systems in the subsystem energy eigenbasis using modular functions. QDT requires no user knowledge of operator matrix assembly and automatically performs all necessary operator constructions and Hilbert space expansions. Density matrix propagation is performed by numerically solving the Liouville-von-Neumann equation. In order to simulate dissipation and decoherence effects, the Lindblad formalism is implemented. Furthermore, QDT supplies practical analysis and plotting functions, such as visualization of density matrix and expectation value dynamics, that facilitate the evaluation of simulation results. QDT further provides a module for the simulation of coherent multidimensional spectroscopy.The covariance extension equation: a Riccati-type approach to analytic interpolationhttps://zbmath.org/1541.930382024-09-27T17:47:02.548271Z"Cui, Yufang"https://zbmath.org/authors/?q=ai:cui.yufang"Lindquist, Anders"https://zbmath.org/authors/?q=ai:lindquist.anders-gEditorial remark: No review copy delivered.Equivalent diagrams of fractional order elementshttps://zbmath.org/1541.932332024-09-27T17:47:02.548271Z"Różowicz, Sebastian"https://zbmath.org/authors/?q=ai:rozowicz.sebastian.1"Włodarczyk, Maciej"https://zbmath.org/authors/?q=ai:wlodarczyk.maciej"Zawadzki, Andrzej"https://zbmath.org/authors/?q=ai:zawadzki.andrzejSummary: This paper presents equivalent impedance and operator admittance systems for fractional order elements. Presented models of fractional order elements of the type: \(s^\alpha L_\alpha\) and \(1/s^\alpha C_\alpha\), \((0\alpha 1)\) were obtained using the Laplace transform based on the expansion of the factor sign to an infinite fraction with varying degrees of accuracy -- the continued fraction expansion method (CFE). Then circuit synthesis methods were applied. As a result, equivalent circuit diagrams of fractional order elements were obtained. The obtained equivalent schemes consist both of classical RLC elements, as well as active elements built based on operational amplifiers. Numerical experiments were conducted for the constructed models, presenting responses to selected input signals.Sampling numbers of smoothness classes via \(\ell^1\)-minimizationhttps://zbmath.org/1541.940342024-09-27T17:47:02.548271Z"Jahn, Thomas"https://zbmath.org/authors/?q=ai:jahn.thomas.1"Ullrich, Tino"https://zbmath.org/authors/?q=ai:ullrich.tino"Voigtlaender, Felix"https://zbmath.org/authors/?q=ai:voigtlaender.felixSummary: Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in \(L^2\). In particular, we show that in relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in \(L^2\) can be upper bounded by best \(n\)-term trigonometric widths in \(L^\infty\). We describe a recovery procedure from \(m\) function values based on \(\ell^1\)-minimization (basis pursuit denoising). With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of \(m^{- 1/2}\) (up to log factors) compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to \(S_p^r W(\mathbb{T}^d)\) on the \(d\)-torus with a logarithmically better rate of convergence than any linear method can achieve when \(1 < p < 2\) and \(d\) is large. This effect is not present for isotropic Sobolev spaces.