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This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit (pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on sinc matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the sinc matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation. [ABSTRACT FROM AUTHOR]

We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data. [ABSTRACT FROM AUTHOR]

We design an Algorithm to fabricate universal holomorphic maps between any two complex Euclidean spaces, within preassigned transcendental growth rate. As by-products, universal holomorphic maps from C n to C P m ( n ⩽ m ) and to complex tori having slow growth are obtained. We take inspiration from Oka manifolds theory, Nevanlinna theory, and hypercyclic operators theory. [ABSTRACT FROM AUTHOR]

Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space H 2 , a new H 2 -like optimal model reduction problem is introduced and first-order optimality conditions are derived. As in the classical H 2 case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method. [ABSTRACT FROM AUTHOR]

We investigate asymptotic behavior of polynomials Q n (z) satisfying non-Hermitian orthogonality relations ∫ Δ s k Q n (s) ρ (s) d s = 0 , k ∈ { 0 , ... , n - 1 } , where Δ is a Chebotarëv (minimal capacity) contour connecting three non-collinear points and ρ (s) is a Jacobi-type weight including a possible power-type singularity at the Chebotarëv center of Δ . [ABSTRACT FROM AUTHOR]

In various applications, the adoption of optimal energy matching pursuit with dictionary elements is common. When the dictionary elements are indexed by parameters within a bounded region, exhaustion-type algorithms can be employed. This article aims to investigate a process that converts the optimal parameter selection in unbounded regions to a bounded and closed (compact) sub-domain. Such a process provides accessibility for energy matching pursuit in a wide range of applications. The paper initially focuses on the open unit disc and the upper-half complex plane, introducing adaptive Fourier decomposition as the underlying methodology. It then extends this concept to general Hilbert spaces with a dictionary and Bochner spaces for random signals. Computational examples are included to illustrate the concepts presented. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Jackson type theorem of approximation by truncated max-product sampling Kantorovich operators in L p spaces. Our results generalize those of Coroianu et al. (Anal Appl 19:219–244, 2021) and Coroianu and Gal (J Integral Equat Appl 29:349–364, 2017). We use the equivalence between the K - functional and the modulus of continuity of f, and the Hardy–Littlewood maximal function as the main tools in the proofs, and also give some examples to apply the main result. [ABSTRACT FROM AUTHOR]

Let g n , n = 1 , 2 , ... , be the logarithmic derivative of a complex polynomial having all zeros on the unit circle, i.e., a function of the form g n (z) = (z - z 1) - 1 + ⋯ + (z - z n) - 1 , | z 1 | = ⋯ = | z n | = 1 . For any p > 0 , we establish the bound ∫ - 1 1 | g n (x) | p d x > C p n p - 1 , sharp in the order of the quantity n, where C p > 0 is a constant, depending only on p. The particular case p = 1 of this inequality can be considered as a stronger variant of the well-known estimate ∬ | z | < 1 | g n (z) | d x d y > c > 0 for the area integral of g n , obtained by Newman (Am Math Mon 79(9):1015–1016, 1972). The result also shows that the set { g n } is not dense in the spaces L p [ - 1 , 1 ] , p ≥ 1 . [ABSTRACT FROM AUTHOR]

We propose a strategy for greedy sampling in the context of non-intrusive interpolation-based surrogate modeling for frequency-domain problems. We rely on a non-intrusive and cheap error indicator to drive the adaptive selection of the high-fidelity samples on which the surrogate is based. We develop a theoretical framework to support our proposed indicator. We also present several practical approaches for the termination criterion that is used to end the greedy sampling iterations. To showcase our greedy strategy, we numerically test it in combination with the well-known Loewner framework. To this effect, we consider several benchmarks, highlighting the effectiveness of our adaptive approach in approximating the transfer function of complex systems from a few samples. [ABSTRACT FROM AUTHOR]

In various fields of science and engineering, such as financial mathematics, mathematical physics models, and radiation transfer, stochastic integral equations are important and practical tools for modeling and describing problems. Due to the existence of random factors, we face a fundamental problem in solving stochastic integral equations, and that is the lack of analytical solutions or the great complexity of these solutions. Therefore, finding an efficient numerical solution is essential. In this paper, we intend to propose and study a new method based on the Floater-Hormann interpolation and the spectral collocation method for linear and nonlinear stochastic Itô-Volterra integral equations (SVIEs). The Floater-Hormann interpolation offers an approximation regardless of the distribution of the points. Therefore, this method can be mentioned as a meshless method. The presented method reduces SVIEs under consideration into a system of algebraic equations that can be solved by the appropriate method. We presented an error bound to be sure of the convergence and reliability of the method. Finally, the efficiency and the applicability of the present scheme are investigated through some numerical experiments. [ABSTRACT FROM AUTHOR]

Let f be an analytic function on a simply-connected compact continuum E of the complex z-plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points z ∈ C ? How can we estimate the locations of real or complex singularities of f? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs. [ABSTRACT FROM AUTHOR]

We construct explicit universal entire curves h : C → C P n whose Nevanlinna characteristic functions grow slower than any preassigned transcendental growth rate. Moreover, we can make h to be hypercyclic for translation operations along any given countable directions. [ABSTRACT FROM AUTHOR]

In this paper we compare three different orthogonal systems in L 2 (R) which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrödinger equation on the real line, specifically, stretched Fourier functions, Hermite functions and Malmquist–Takenaka functions. All three have banded skew-Hermitian differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary—this is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the L 2 (R) norm. We derive asymptotic approximations of the coefficients for a wave packet in each of these bases, which are extremely accurate in the high frequency regime. We show that the Malmquist–Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions and stretched Fourier expansions for approximating wave packets. [ABSTRACT FROM AUTHOR]

We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced samples. Although it is always better to approximate from large numbers of samples if they are available, whether equispaced or not, this method often performs impressively even when the sampling grid is coarse. In most cases it gives more accurate approximations than other methods. We support this claim with a review and discussion of nine classes of existing methods in the light of general properties of approximation theory as well as the “impossibility theorem” for equispaced approximation. We make careful use of numerical experiments, which are summarized in a sequence of nine figures. Among our new contributions is the observation, summarized in Fig. 7, that methods such as polynomial least-squares and Fourier extension may be either exponentially accurate and exponentially unstable, or less accurate and stable, depending on implementation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce three series of Jacobi rational basis functions on the half line by using the matrix decomposition technique. The new basis functions are simultaneously orthogonal in both L 2 - and H 1 -inner products, and lead to diagonal systems for second order problems with constant coefficients. We construct efficient space-time spectral methods for parabolic problems on semi-infinite domains using Jacobi rational approximation in space and multi-domain Legendre–Gauss collocation approximation in time, which can be implemented in a synchronous parallel fashion. Some rigorous error estimates are carried out for one-dimensional parabolic equations. Numerical results demonstrate that the suggested approaches possess high-order accuracy and greatly improve the efficiency. [ABSTRACT FROM AUTHOR]

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case. [ABSTRACT FROM AUTHOR]

The barycentric forms of polynomial and rational interpolation have recently gained popularity, because they can be computed with simple, efficient, and numerically stable algorithms. In this paper, we show more generally that the evaluation of any function that can be expressed as r (x) = ∑ i = 0 n a i (x) f i / ∑ j = 0 m b j (x) in terms of data values f i and some functions a i and b j for i = 0 , ... , n and j = 0 , ⋯ , m with a simple algorithm that first sums up the terms in the numerator and the denominator, followed by a final division, is forward and backward stable under certain assumptions. This result includes the two barycentric forms of rational interpolation as special cases. Our analysis further reveals that the stability of the second barycentric form depends on the Lebesgue constant associated with the interpolation nodes, which typically grows with n, whereas the stability of the first barycentric form depends on a similar, but different quantity, that can be bounded in terms of the mesh ratio, regardless of n. We support our theoretical results with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose two kinds of extended barycentric rational schemes via (non-conformally) scaled transformations for approximating functions of singularities, which are bulit upon applying the barycentric interpolation formula of the second kind at two kinds of mapped nodes: (i) equispaced nodes and (ii) (shifted) Chebyshev nodes. While the weights in interpolation formula are selected inspired by the works of Berrut, Floater and Hormann. Ample numerical tests show that the extended barycentric rational schemes are efficient and can achieve higher convergence rates as the scaled parameter and the degree of the local approximation polynomial increase. Moreover, from the barycentric formula, it is easy to derive the difference matrices at these mapped nodes, which leads to an accurate Levin method for dealing with highly oscillatory integrals with the integrands of algebraic singularities. Numerical experiments are carried out to illustrate the effectiveness and accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, for the neural network (NN) operators activated by sigmoidal functions, we will obtain quantitative estimates in direct connection with the asymptotic behavior of their activation functions. We will cover all cases of discrete and Kantorovich type versions of NN operators in both univariate and multivariate settings since the proofs of the latter ones can be reduced in a convenient way to the proofs in the univariate cases. The above mentioned quantitative estimates have been achieved for continuous functions and using the well-known modulus of continuity. Further, in the case of the Kantorovich version of the NN operators, we also provide quantitative estimates for L p functions with respect to the L p norm; this has been done using the well-known Peetre K-functionals. At the end of the paper, some examples of sigmoidal functions are presented; also the case of NN operators activated by the rectified linear unit (ReLU) functions are considered. Furthermore, numerical examples in both univariate and multivariate cases have been provided in order to show the approximation performances of all the above operators. [ABSTRACT FROM AUTHOR]

We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error 1 − r/f on the spectral interval of A. By minimizing this upper bound over all interpolation points, we obtain a new, simple, and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant r. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating r(A), where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for r following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of r leading to a small error, even in presence of finite precision arithmetic. [ABSTRACT FROM AUTHOR]

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of constructioning such approximations using polynomials in the Bernstein basis. We consider a family of inequality-constrained quadratic programs. In the univariate case, a quadratic cone constraint allows us to search over all nonnegative polynomials of a given degree. In both the univariate and multivariate cases, we consider approximate problems with linear inequality constraints. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate 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 are obtained on rational curves. [ABSTRACT FROM AUTHOR]

The power series expansions of the hypergeometric functions p + 1 F p (a , b 1 , ... , b p ; c 1 , ... , c p ; z) converge either inside the unit disk | z | < 1 or outside this disk | z | > 1 . Nørlund's expansion in powers of z / (z - 1) converges in the half-plane R (z) < 1 / 2 . For arbitrary z 0 ∈ C , Bühring's expansion in inverse powers of z - z 0 converges outside the disk | z - z 0 | = max { | z 0 | , | z 0 - 1 | } . None of them converge on the whole indented closed unit disk | z | ≤ 1 , z ≠ 1 . In this paper, we derive new expansions in terms of rational functions of z that converge in different regions, bounded or unbounded, of the complex plane that contain the indented closed unit disk. We give either explicit formulas for the coefficients of the expansions or recurrence relations. The key point of the analysis is the use of multi-point Taylor expansions in appropriate integral representations of p + 1 F p (a , b 1 , ... , b p ; c 1 , ... , c p ; z) . We show the accuracy of the approximations by means of several numerical experiments. [ABSTRACT FROM AUTHOR]

We establish an equivalence between two classes of methods for solving fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational Krylov Methods (RKM). In particular, we demonstrate that several recently proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed point of view allows us to give convergence proofs for some methods where none were previously available. We also propose a new RKM for fractional diffusion problems with poles chosen using the best rational approximation of the function z−s with z ranging over the spectral interval of the spatial discretization matrix. We prove convergence rates for this method and demonstrate numerically that it is competitive with or superior to many methods from the reduced basis, rational Krylov, and direct rational approximation classes. We provide numerical tests for some elliptic fractional diffusion model problems. [ABSTRACT FROM AUTHOR]

The Floater--Hormann family of rational interpolants do not have spuriouspoles or unattainable points, are efficient to calculate, and have arbitrarilyhigh approximation orders. One concern when using them is that theamplification of rounding errors increases with approximation order, and canmake balancing the interpolation error and rounding error difficult. Thisarticle proposes to modify the Floater--Hormann interpolants by includingadditional local polynomial interpolants at the ends of the interval. Thisappears to improve the conditioning of the interpolants and allow higherapproximation orders to be used in practice.

Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for approximating any continuous function, and the convergence rate of the interpolants approximating a function is obtained using the modulus of continuity. [ABSTRACT FROM AUTHOR]

We consider vector-valued Banach function algebras on a compact Hausdorff space. Then, we define the subalgebras generated by vector-valued polynomials and rational functions, and determine their maximal ideal spaces and Šilov boundaries. We finally make use the results for a certain category of these algebras such as vector-valued Lipschitz algebras, vector-valued Dales–Davie algebras (algebras of vector-valued differentiable functions) and the algebras of vector-valued differentiable Lipschitz functions. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop the diagonalized Legendre rational spectral method for exterior problems. We first consider the exterior problems of two-dimensional elliptic and parabolic equations in polar coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. Then we consider the exterior problems of three-dimensional elliptic and parabolic equations in spherical coordinates, construct the Sobolev orthogonal Legendre rational basis functions, and then propose the diagonalized Legendre rational spectral methods. The main advantages of the suggested approaches are that the discrete systems are diagonal and the numerical solutions can be represented as truncated Fourier series. The numerical results show their effectiveness and accuracy. [ABSTRACT FROM AUTHOR]

In the field of model order reduction for frequency response problems, the minimal rational interpolation (MRI) method has been shown to be quite effective. However, in some cases, numerical instabilities may arise when applying MRI to build a surrogate model over a large frequency range, spanning several orders of magnitude. We propose a strategy to overcome these instabilities, replacing an unstable global MRI surrogate with a union of stable local rational models. The partitioning of the frequency range into local frequency sub-ranges is performed automatically and adaptively, and is complemented by a (greedy) adaptive selection of the sampled frequencies over each sub-range. We verify the effectiveness of our proposed method with two numerical examples. [ABSTRACT FROM AUTHOR]

Based on the blossoming theory, in this work we develop a new method for deriving Hermite-Padé approximants of certain hypergeometric series. Its general principle consists in building identities generalising the Hermite identity for exponentials, and in then applying their blossomed versions to appropriate tuples to simultaneously produce explicit expressions of the approximants and explicit integral representations of the corresponding remainders. For binomial series we use classical blossoms while for q-hypergeometric series we have to use q-blossoms. [ABSTRACT FROM AUTHOR]

A new mathematical formula was derived for near equilibrium relaxation processes of enzyme reactions including the conformational selection (CS) modes. CS is one of the most accepted molecular recognition mechanisms, in which protein conformers (CS conformers) are in an equilibrium with varying degrees of ligand binding affinity so that ligands select a particular conformer among them to bind. Using computer simulation techniques, our previous study (Egawa and Callender in Math Biosci 313: 61–70, 2019) predicted that the rate constant for the near equilibrium relaxation processes (kNER) and the concentration-sum of substrate and product (CLt) of enzyme reactions uniquely related to the presence of CS steps in a manner that 1/kNER versus CLt plot transformed from linear to quadratic as the elementary rate constants of inter-conversions among the CS conformers were becoming smaller relative to the rate constants at other steps of the enzymatic reaction system. Thus our previous work could provide a potential tool to detect the presence of CS steps in an enzyme reaction simply by assays using only trace amount of enzyme samples, although logical basis to have the quadratic deformation in the 1/kNER versus CLt plot in response to the presence of CS steps could not be clarified. Employing mathematical approaches that were alternative to those in our previous study, this study succeeded in deriving a theoretical equation that fully explained why and how the CS modes caused the quadratic characters in the 1/kNER versus CLt plot. [ABSTRACT FROM AUTHOR]