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A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments. [ABSTRACT FROM AUTHOR]

This paper generates a novel approach called the clique polynomial method (CPM) using the clique polynomials raised in graph theory and used for solving the fractional order PDE. The fractional derivative is defined in terms of the Caputo fractional sense and the fractional partial differential equations (FPDE) are converted into nonlinear algebraic equations and collocated with suitable grid points in the current approach. The convergence analysis for the proposed scheme is constructed and the technique proved to be uniformly convegant. We applied the method for solving four problems to justify the proposed technique. Tables and graphs reveal that this new approach yield better results. Some theorems are discussed with proof. [ABSTRACT FROM AUTHOR]

The best polynomial approximation and the Chebyshev approximation are both important in numerical analysis. In tradition, the best approximation is regarded as better than the Chebyshev approximation, because it is usually considered in the uniform norm. However, it is not always superior to the latter, as noticed by Trefethen (Math. Today 47:184–188, 2011) for the function f (x) = | x - 0.25 | . Recently, Wang (arxiv, 2106.03456, 2021) observed a similar phenomenon for a function with an algebraic singularity and proved it in theory. In this paper, we find that for functions with logarithmic endpoint singularities, the pointwise errors of the Chebyshev approximation are smaller than those of the best approximations of the same degree, except only at the narrow boundary layer. The pointwise error for the Chebyshev series truncated at the degree n is O (n - κ) ( κ = min { 2 γ + 1 , 2 δ + 1 } ), but increases as O (n - (κ - 1)) at the endpoint singularities. Theorems are given to explain this effect. [ABSTRACT FROM AUTHOR]

In the given note, we present the Durrmeyer modification of Lupaş type operators having Pochhammer k - symbol. We study convergence, asymptotic result and error of approximation in terms of first order modulus of continuity for these operators. Also, we estimate the approximation behavior of these operators in two variables in terms of complete and partial modulus of continuity. At last, we unveil some numerical examples with graphs to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Kantorovich operators on Morrey spaces. We first recall the main tool in the proof of the uniform boundedness of Kantorovich operators in Morrey spaces, namely a pointwise estimate of Kantorovich operators by the Hardy–Littlewood maximal operator. We also investigate the convergence of Kantorovich operators in Morrey spaces under weaker assumptions that is also true for the endpoint case. In addition, we obtain the rate of convergence of Kantorovich operators in Morrey spaces under the condition of Hölder continuity. Our result can be applicable to the function spaces in a recent result by Zeren, Ismailov and Karacam. [ABSTRACT FROM AUTHOR]

In this paper we consider a circular arc approximation by hexic polynomial curves having 12 contacts with the circular arc. We present two methods for obtaining G k approximation curves, k = 3 , 4 , which interpolate at both endpoints and the midpoint of the circular arc. The approximation curves can be obtained by solving an equation of degree six. We show that the approximation orders of our methods are 12. We find the optimal approximation for each method and present numerical examples illustrating that the approximation orders are 12. [ABSTRACT FROM AUTHOR]

Given pointwise samples of an unknown function belonging to a certain model set, one seeks in optimal recovery to recover this function in a way that minimizes the worst-case error of the recovery procedure. While it is often known that such an optimal recovery procedure can be chosen to be linear, e.g., when the model set is based on approximability by a subspace of continuous functions, a construction of the procedure is rarely available. This note uncovers a practical algorithm to construct a linear optimal recovery map when the approximation space is a Chevyshev space of univariate functions that has dimension at least three and contains the constants. [ABSTRACT FROM AUTHOR]

The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered. [ABSTRACT FROM AUTHOR]

In this paper we present a new perspective on error analysis for Legendre approximations of differentiable functions. We start by introducing a sequence of Legendre–Gauss–Lobatto polynomials and prove their theoretical properties, including an explicit and optimal upper bound. We then apply these properties to derive a new explicit bound for the Legendre coefficients of differentiable functions. Building on this, we establish an explicit and optimal error bound for Legendre approximations in the L 2 norm and an explicit and optimal error bound for Legendre approximations in the L ∞ norm under the condition that their maximum error is attained in the interior of the interval. Illustrative examples are provided to demonstrate the sharpness of our new results. [ABSTRACT FROM AUTHOR]

We construct and study sequences of linear operators of Bernstein-type acting on bivariate functions defined on the unit disk. To this end, we study Bernstein-type operators under a domain transformation, we analyze the bivariate Bernstein–Stancu operators, and we introduce Bernstein-type operators on disk quadrants by means of continuously differentiable transformations of the function. We state convergence results for continuous functions and we estimate the rate of convergence. Finally some interesting numerical examples are given, comparing approximations using the shifted Bernstein–Stancu and the Bernstein-type operator on disk quadrants. [ABSTRACT FROM AUTHOR]

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Łojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition), this Łojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre's Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo-moment sequences, which are positive on the quadratic module of S. [ABSTRACT FROM AUTHOR]

We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Importantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices. [ABSTRACT FROM AUTHOR]

In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points inside a leaf box in the fast multipole method (FMM) hierarchical tree structure. The QB2X method explicitly includes nonlinear dependency of the boundary geometry in the plane wave expansions, thereby providing for higher-order representations of both the boundary geometry and density functions in the integrand, with its convergence following standard FMM error analysis. Numerical results are presented to demonstrate the performance of the QB2X method for Helmholtz layer potentials using one expansion center for the entire FMM-leaf box for both flat and curved boundaries with various densities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Müntz-Legendre polynomial, which is generated from the fractional basis, and develop a collocation method with the help of operational matrices to solve the stochastic fractional integro-differential equation (SFIDE). For the need of our numerical method, we introduce the operational matrices of the Müntz-Legendre polynomial for the operators involved in the SFIDE. Moreover, we proposed an algorithm with the help of the differential evolution algorithm to find the Müntz space on which we get the best results. Further, we study the convergence analysis of the proposed method. Finally, the accuracy and efficiency of the numerical method show through some test examples. [ABSTRACT FROM AUTHOR]

We prove various theorems on approximation using polynomials with integer coefficients in the Bernstein basis of any given order. In the extreme, we draw the coefficients from { ± 1 } only. A basic case of our results states that for any Lipschitz function f : [ 0 , 1 ] → [ - 1 , 1 ] and for any positive integer n, there are signs σ 0 , ⋯ , σ n ∈ { ± 1 } such that f (x) - ∑ k = 0 n σ k n k x k (1 - x) n - k ≤ C (1 + | f | Lip) 1 + n x (1 - x) for all x ∈ [ 0 , 1 ]. More generally, we show that higher accuracy is achievable for smoother functions: For any integer s ≥ 1 , if f has a Lipschitz (s - 1) st derivative, then approximation accuracy of order O (n - s / 2) is achievable with coefficients in { ± 1 } provided ‖ f ‖ ∞ < 1 , and of order O (n - s) with unrestricted integer coefficients, both uniformly on closed subintervals of (0, 1) as above. Hence these polynomial approximations are not constrained by the saturation of classical Bernstein polynomials. Our approximations are constructive and can be implemented using feedforward neural networks whose weights are chosen from { ± 1 } only. [ABSTRACT FROM AUTHOR]

We consider approximating analytic functions on the interval [ - 1 , 1 ] from their values at a set of m + 1 equispaced nodes. A result of Platte, Trefethen & Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this 'impossibility' theorem. Our 'possibility' theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance ϵ > 0 , which in practice can be chosen close to machine epsilon. The method is known as polynomial frame approximation or polynomial extensions. It uses algebraic polynomials of degree n on an extended interval [ - γ , γ ] , γ > 1 , to construct an approximation on [ - 1 , 1 ] via a SVD-regularized least-squares fit. A key step in the proof of our main theorem is a new result on the maximal behaviour of a polynomial of degree n on [ - 1 , 1 ] that is simultaneously bounded by one at a set of m + 1 equispaced nodes in [ - 1 , 1 ] and 1 / ϵ on the extended interval [ - γ , γ ] . We show that linear oversampling, i.e. m = c n log (1 / ϵ) / γ 2 - 1 , is sufficient for uniform boundedness of any such polynomial on [ - 1 , 1 ] . This result aside, we also prove an extended impossibility theorem, which shows that such a possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal. [ABSTRACT FROM AUTHOR]

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in Xu (J Funct Anal 281(12):109257, 2021) for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed-form formulas for the reproducing kernels. The main results provide construction of semi-discrete localized tight frame in weighted L 2 norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain. [ABSTRACT FROM AUTHOR]

We introduce the notion of metric divided differences of set-valued functions. With this notion we obtain bounds on the error in set-valued metric polynomial interpolation. These error bounds lead to high-order approximations of set-valued functions by metric piecewise-polynomial interpolants of high degree. Moreover, we derive high-order approximation of set-valued functions by local metric approximation operators reproducing high-degree polynomials. [ABSTRACT FROM AUTHOR]

The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems with a regularize-first approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods. Furthermore, the difficult process of explicitly choosing a suitable discretization is not required. [ABSTRACT FROM AUTHOR]

We present a new time-adaptive FMM for a space-time boundary element method for the heat equation. The method extends the existing parabolic FMM by adding new operations that allow for an efficient treatment of tensor product meshes which are adaptive in time. We analyze the efficiency of the new operations and the approximation quality of the related kernel expansions and present numerical experiments that reveal the benefits of the new method. [ABSTRACT FROM AUTHOR]

In this work, we study a global quadrature scheme for analytic functions on compact intervals based on function values on quasi-uniform grids of quadrature nodes. In practice it is not always possible to sample functions at optimal nodes that give well-conditioned and quickly converging interpolatory quadrature rules at the same time. Therefore, we go beyond classical interpolatory quadrature by lowering the degree of the polynomial approximant and by applying auxiliary mapping functions that map the original quadrature nodes to more suitable fake nodes. More precisely, we investigate the combination of the Kosloff Tal-Ezer map and least-squares approximation (KTL) for numerical quadrature: a careful selection of the mapping parameter ensures stability of the scheme, a high accuracy of the approximation and, at the same time, an asymptotically optimal ratio between the degree of the polynomial and the spacing of the grid. We will investigate the properties of this KTL quadrature and focus on the symmetry of the quadrature weights, the limit relations for the mapping parameter, as well as the computation of the quadrature weights in the standard monomial and in the Chebyshev bases with help of a cosine transform. Numerical tests on equispaced nodes show that a static choice of the map’s parameter improve the results of the composite trapezoidal rule, while a dynamic approach achieves larger stability and faster convergence, even when the sampling nodes are perturbed. From a computational point of view the proposed method is practical and can be implemented in a simple and efficient way. [ABSTRACT FROM AUTHOR]

Digitized handwritten documents have been used for various purposes, including age detection, a crucial area of research in fields like forensic investigation and medical diagnosis. Automated age recognition is deemed to be a difficult task, due to the great degree of similarity and overlap across people's handwriting. Consequently, the efficiency of the classification system is determined by extracting pertinent features from handwritten documents. This research proposes a set of age-related features suggested by a graphologist to classify handwritten documents into two age groups: youth adult and mature adult. The extracted features are: slant irregularity (SI), pen pressure irregularity (PPI), text lines irregularity (TLI) and the percentage of black and white pixels (PWB). Support Vector Machines (SVM) and Neural Network (NN) classifiers have been used to train, validate and test the proposed approach using two different datasets: the FSHS and the Khatt datasets. When applied to the FSHS dtaset using SVM and NN approaches, the proposed method resulted in a classification accuracy of 71% and 63.5%, respectively. Meanwhile, when applied to the Khatt dataset, our method outperformed state-of-the-art methods with a classification accuracy of 65.2% and 67% utilising SVM and NN classifiers, respectively. These are the best rates available right now in this field. [ABSTRACT FROM AUTHOR]

This paper deals with weighted Lupaş post-quantum Bernstein blending functions and Bézier curves constructed with the help of bases via (p,q)\left(p,q)-integers. These blending functions form normalized totally positive bases. Due to the rational nature of weighted Lupaş post-quantum Bézier curves and positive weights, they help in investigating from geometric point of view. Their degree elevation properties and de Casteljau algorithm have been studied. It has been shown that quadratic weighted Lupaş post-quantum Bézier curves can represent conic sections in two-dimensional plane. Graphical analysis has been presented to discuss geometric interpretation of weight and conic section representation by weighted Lupaş post-quantum Bézier curves. This new generalized weighted Lupaş post-quantum Bézier curve provides better approximation and flexibility to a particular control point as well as control polygon due to extra parameter pp and qq in comparison to classical rational Bézier curves, Lupaş qq-Bézier curves and weighted Lupaş qq-Bézier curves.

The main goal of this article is to present the notion of Fibonacci I-convergence of sequences on intuitionistic fuzzy normed linear space. To accomplish this goal, we mainly investigate some fundamental properties of the newly introduced notion. Then, we examine the Fibonacci I-Cauchy sequences and Fibonacci I completeness in the construction of an intuitionistic fuzzy normed linear space. Some intuitionistic fuzzy Fibonacci ideal convergent spaces have been established. Further, we prove on some algebraic and topological features of these convergent sequence spaces.

Our aim is to define a new mean having specific properties which are crucial in several applications. As examples of this kind of applications, we can mention subdivision and multiresolution schemes based on nonuniform meshes and the numerical solution of conservation laws. Specifically, we pay attention to three relevant properties: to remain close to the minimum of the two values, to give a closer value to the weighted arithmetic mean of the two values when they are already similar, and to accomplish a Lipchitz condition. We define a family of means depending on a parameter k that matches these requirements. Using these means, it is possible to obtain more accurate theoretical constants in some stability theorems, and therefore they are better fitted to what happens in real examples. We carry out examples in the fields of subdivision schemes and hyperbolic conservation laws. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce and investigate a new d-orthogonal family generalizing the Laguerre polynomial set. We use operational rules associated with corresponding lowering, raising and transfer operators to derive many related properties, namely, recurrence relation, d-orthogonality, functional vector of d-orthogonality, inversion and connection relations, differential equation, among others. [ABSTRACT FROM AUTHOR]

In the present paper we show that in Pólya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a natural convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the error of approximation for Bernstein–Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proofs rely on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first moment of Pólya random variables with respect to the replacement parameter. [ABSTRACT FROM AUTHOR]

In the present paper, we have proposed the Stancu type generalization of the family of Bernstein Kantorovich operators involving parameter α ∈ [ 0 , 1 ] with Shifted Knots. These operators provide the pliability to approximate on the interval [0,1] and over its sub intervals. For the proposed operators, we investigate some basic results of approximation and their rate of convergence in terms of first and second order modulus of continuity, Lipschitz class and Lipschitz-type function. We also estimate the global rate of convergence of the operators with the help of Ditzian-Totik modulus of smoothness. Moreover, the pth order generalization of the operators is established. Some numerical simulation and graphical comparisons are given for better depiction of theoretical results. [ABSTRACT FROM AUTHOR]

We consider the problem of the uniform (in L ∞ ) recovery of ridge functions f (x) = φ (⟨ a , x ⟩) , x ∈ B 2 n , using noisy evaluations y 1 ≈ f (x 1) , ... , y N ≈ f (x N) . It is known that for classes of functions φ of finite smoothness the problem suffers from the curse of dimensionality: in order to provide good accuracy for the recovery it is necessary to make exponential number of evaluations. We prove that if φ is analytic in a neighborhood of [ - 1 , 1 ] and the noise is very small, ε ⩽ exp (- c log 2 n) , then there is an efficient algorithm that recovers f with good accuracy using O (n log 2 n) function evaluations. [ABSTRACT FROM AUTHOR]

We introduce a method for the nonlinear dimension reduction of a high-dimensional function |$u:{\mathbb{R}}^d\rightarrow{\mathbb{R}}$|⁠ , |$d\gg 1$|⁠. Our objective is to identify a nonlinear feature map |$g:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^m$|⁠ , with a prescribed intermediate dimension |$m\ll d$|⁠ , so that |$u$| can be well approximated by |$f\circ g$| for some profile function |$f:{\mathbb{R}}^m\rightarrow{\mathbb{R}}$|⁠. We propose to build the feature map by aligning the Jacobian |$\nabla g$| with the gradient |$\nabla u$|⁠ , and we theoretically analyze the properties of the resulting |$g$|⁠. Once |$g$| is built, we construct |$f$| by solving a gradient-enhanced least squares problem. Our practical algorithm uses a sample |$\{{\textbf{x}}^{(i)},u({\textbf{x}}^{(i)}),\nabla u({\textbf{x}}^{(i)})\}_{i=1}^N$| and builds both |$g$| and |$f$| on adaptive downward-closed polynomial spaces, using cross validation to avoid overfitting. We numerically evaluate the performance of our algorithm across different benchmarks, and explore the impact of the intermediate dimension |$m$|⁠. We show that building a nonlinear feature map |$g$| can permit more accurate approximation of |$u$| than a linear |$g$|⁠ , for the same input data set. [ABSTRACT FROM AUTHOR]

This article introduces the novel notion of dimension preserving approximation for continuous functions defined on [0, 1] and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated. [ABSTRACT FROM AUTHOR]

This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to n + k with 0 < k ≤ n . Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the L 2 norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as n → ∞ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs. [ABSTRACT FROM AUTHOR]

We investigate the computation and stability of hierarchical Newton interpolation formulas using ℜ -Leja sequences. We carry out a thorough study, in particular from the angle of rounding errors (underflow/overflow), cost, and numerical stability. Furthermore, conditionings of Newton formulas are addressed and moderate estimates are derived. [ABSTRACT FROM AUTHOR]

Given gridded cell-average data of a smooth multivariate function, we present a constructive explicit procedure for generating a high-order global approximation of the function. One contribution is the derivation of high-order approximations to point-values of the function directly from the cell-average data. The second contribution is the development of univariate B-spline-based high-order quasi-interpolation operators using cell-average data. Multivariate spline quasi-interpolation approximation operators are obtained by tensor product of the univariate operators. [ABSTRACT FROM AUTHOR]

We call a hybrid polynomial a function Q (x) = α x 2 + β x + T (x) , where α , β ∈ R and T is a trigonometric polynomial. For s ≥ 1 , let Y s : = { y i } i ∈ Z , be such that y i + 2 π = y i + 2 s , i ∈ Z . We approximate a function f (x) : = g (x) + α x 2 , where g is a continuous 2 π -periodic function and α ∈ R , and f has the collection Y s as its set of inflection points, by hybrid polynomials having the same inflection points. The constant α makes the problem much different than ordinary coconvex trigonometric approximation. [ABSTRACT FROM AUTHOR]