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The implicit boundary integral method (IBIM) provides a framework to construct quadrature rules on regular lattices for integrals over irregular domain boundaries. This work provides a systematic error analysis for IBIMs on uniform Cartesian grids for boundaries with different degrees of regularity. First, it is shown that the quadrature error gains an additional order of d - 1 2 from the curvature for a strongly convex smooth boundary due to the “randomness” in the signed distances. This gain is discounted for degenerated convex surfaces. Then the extension of error estimate to general boundaries under some special circumstances is considered, including how quadrature error depends on the boundary’s local geometry relative to the underlying grid. Bounds on the variance of the quadrature error under random shifts and rotations of the lattices are also derived. [ABSTRACT FROM AUTHOR]

In the paper, the authors establish several integral inequalities of the Ostrowski type for s-logarithmically convex functions. These integral inequalities modify the conditions and correct errors in two main theorems of the paper [A. O. Akdemir and M. Tunç, Ostrowski type inequalities for s-logarithmically convex functions in the second sense with applications, Georgian Math. J.22 (2015), no. 1, 1–7]. [ABSTRACT FROM AUTHOR]

We give investigations on the approximation order of translation networks produced by the convolution translation operators defined on a Jacobi cone and the surface cone. We deal with the convolution translation from the view of Fourier analysis, express the translation operator with orthogonal basis and provide a sufficient condition to ensure the density for the translation networks. Based on these facts, we construct with the near best approximation operator and the Gauss integral formula two kinds of translation network operators and show their approximation orders in the best polynomial approximation. [ABSTRACT FROM AUTHOR]

Given a sequence of Marcinkiewicz–Zygmund inequalities in L 2 on a compact space, Gröchenig (J Approx Theory 257:105455, 2020) discussed weighted least squares approximation and least squares quadrature. Inspired by this work, for all 1 ≤ p ≤ ∞ , we develop weighted least ℓ p approximation induced by a sequence of Marcinkiewicz–Zygmund inequalities in L p on a compact smooth Riemannian manifold M with normalized Riemannian measure (typical examples are the torus and the sphere). In this paper we derive corresponding approximation theorems with the error measured in L q , 1 ≤ q ≤ ∞ , and least quadrature errors for both Sobolev spaces H p r (M) , r > d / p generated by eigenfunctions associated with the Laplace-Beltrami operator and Besov spaces B p , γ r (M) , 0 < γ ≤ ∞ , r > d / p defined by best ”polynomial” approximation. Finally, we discuss the optimality of the obtained results by giving sharp estimates of sampling numbers and optimal quadrature errors for the aforementioned spaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation method without dividing the domain of the integral equation, as usual, into the sub intervals. Existence and uniqueness of solutions are discussed. Convergence rates and error estimates are given for the numerical scheme in both L ∞ -norm and L 2 -norm. Finally, several numerical examples are provided to show that the proposed method is preferable to its classical predecessor and some other existing approaches. [ABSTRACT FROM AUTHOR]

We present an adaptive method of approximate integration of convex (as well as concave) functions based on a certain refinement of the celebrated Hermite–Hadamard inequality. Numerical experiments are performed and the role of harmonic numbers is shown.

This study systematically develops error estimates tailored to a specific set of general quadrature rules that exclusively incorporate first derivatives. Moreover, it introduces refined versions of select generalized Ostrowski’s type inequalities, enhancing their applicability. Through the skillful application of Hayashi’s celebrated inequality to specific functions, the provided proofs underpin these advancements. Notably, this approach extends its utility to approximate integrals of real functions with bounded first derivatives. Remarkably, it employs Newton-Cotes and Gauss-Legendre quadrature rules, bypassing the need for stringent requirements on higher-order derivatives.

In this study, we comment about a wrong proof, at least incomplete, of the closed Newton Cotes error formulas for integration in a closed interval [a,b].\left[a,b]. These error formulas appear as an intuitive generalization of the simple proof for the error formula of the trapezoidal rule, and their proofs present one controversial step, which converts the proofs in mischievous, or at least, this step needs a clear clarification that it is not easy to derive. The correct proof of such formulas comes from a technique based on the Peano kernel.

Given a positive integer d and a 1 , ⋯ , a r a family of vectors in R d , { k 1 a 1 + ⋯ + k r a r : k 1 , ⋯ , k r ∈ Z } ⊂ R d is the so-called lattice generated by the family. In high dimensional integration, prescribed lattices are used for constructing reliable quadrature schemes. The quadrature points are the lattice points lying on the integration domain, typically the unit hypercube [ 0 , 1) d or a rescaled shifted hypercube. It is crucial to have a cost-effective method for enumerating lattice points within such domains. Undeniably, the lack of such fast enumeration procedures hinders the applicability of lattice rules. Existing enumeration procedures exploit intrinsic properties of the lattice at hand, such as periodicity, orthogonality, recurrences, etc. In this paper, we unveil a general-purpose fast lattice enumeration algorithm based on linear programming (named FLE-LP). [ABSTRACT FROM AUTHOR]

In this note we give representations for the remainder in approximations formulas generated by positive linear operators which preserve some functions including linear functions. We show that in these cases the Peano kernel from the integral representation of the remainder has the same sign on the definition domain. Applications to the iterates of some positive linear operators and quadrature formulas are given. [ABSTRACT FROM AUTHOR]

In this paper, the Black–Scholes (B-S) equation to price American options is studied which is governed by a partial differential problem. A nonlinear partial differential equation (PDE) is resulted by applying a penalty approach for this problem. To numerically solve this PDE, an equipped finite difference method with variable step size (VSS) in the time direction is designed and implemented, coupled with a boundary value solver based on a multiple shooting method (MSM) in the spatial direction. It is completely proven that the proposed approximate option prices satisfy the early exercise constraint. Moreover, the convergence of the proposed method is analysed. The extracted numerical results indicate the punctuality and effectiveness of the present scheme comparing the previous works. [ABSTRACT FROM AUTHOR]

Given a , b ∈ R , m = min { a , b } and M = max { a , b } , we consider the orthogonal polynomials associated with nontrivial positive measures ϕ for which s u p p (ϕ) ⊂ (- ∞ , m ] ∪ [ M , ∞) and (x - a) d ϕ (x) = - (x - b) d ϕ (- x + a + b). For this class of measures, formulas in order to compute the moments, as well as formulas for the weights and nodes in the associated Gaussian quadrature rules are provided. We also show that the QD-algorithm can be applied in order to generate new orthogonal polynomials in a simple way. Several examples are given to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

The Clausen functions arise in numerous applications. An efficient summation/integration method for the numerical calculation of these functions of arbitrary order is proposed in this paper. The method is based on a modification of an earlier method and it does not require the construction of the coefficients in the three-term recurrence relation for the corresponding orthogonal polynomials, but only a transformation of orthogonal polynomials from the real line to the positive semiaxis. Numerical experiments are also included. [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]

In analogy to classical spherical t-design points, we introduce the concept of t-design curves on the sphere. This means that the line integral along a t-design curve integrates polynomials of degree t exactly. For low degrees, we construct explicit examples. We also derive lower asymptotic bounds on the lengths of t-design curves. Our main results prove the existence of asymptotically optimal t-design curves in the Euclidean $2$ -sphere and the existence of t-design curves in the d-sphere.

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]

Let assuming that g ∈ C ∞ [ a , b ] such that f(x) is T-periodic, T = b - a , and f (x) ∈ C ∞ (R t) with R t = R \ { t + k T } k = - ∞ ∞ . Here stands for the Hadamard Finite Part (HFP) of the singular integral ∫ a b f (x) d x that does not exist in the regular sense. In a recent work, we unified the treatment of these HFP integrals by using a generalization of the Euler–Maclaurin expansion due to the author and developed a number of numerical quadrature formulas T ^ m , n (s) [ f ] of trapezoidal type for I[f] for all m. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case m = 3 , and these are T ^ 3 , n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , h = T n , T ^ 3 , n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , h = T n , T ^ 3 , n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4) , h = T n. For all m and s, we showed that all of the numerical quadrature formulas T ^ m , n (s) [ f ] have spectral accuracy; that is, T ^ m , n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In this work, we continue our study of convergence and extend it to functions f(x) that possess certain analyticity properties. Specifically, we assume that f(z), as a function of the complex variable z, is also analytic in the infinite strip | Im z | < σ for some σ > 0 , excluding the poles of order m at the points t + k T , k = 0 , ± 1 , ± 2 , …. For m = 1 , 2 , 3 , 4 and relevant s, we prove that T ^ m , n (s) [ f ] - I [ f ] = O (exp (- 2 π n ρ / T)) as n → ∞ ∀ ρ < σ. [ABSTRACT FROM AUTHOR]

This work studies the problem of construction of optimal quadrature formulas in the sense of Sard in the space L 2 (m) (0 , 1) for numerical calculation of Fourier coefficients. Using Sobolev's method, we obtain new sine and cosine weighted optimal quadrature formulas of such type for N + 1 ≥ m , where N + 1 is the number of nodes. Then, explicit formulas for the optimal coefficients of optimal quadrature formulas are obtained. The obtained optimal quadrature formulas in L 2 (m) (0 , 1) space are exact for algebraic polynomials of degree (m − 1) . [ABSTRACT FROM AUTHOR]

We consider the efficient construction of polynomial lattice rules, which are special cases of so-called quasi-Monte Carlo (QMC) rules. These are of particular interest for the approximate computation of multivariate integrals where the dimension d may be in the hundreds or thousands. We study a construction method that assembles the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a digit-by-digit (or, equivalently, coefficient-by-coefficient) fashion. As we will show, the integration error of the corresponding QMC rules achieves excellent convergence order, and, under suitable conditions, we can vanquish the curse of dimensionality by considering function spaces equipped with coordinate weights. The construction algorithm is based on a quality measure that is independent of the underlying smoothness of the function space and can be implemented in a fast manner (without the use of fast Fourier transformations). Furthermore, we illustrate our findings with extensive numerical results. [ABSTRACT FROM AUTHOR]

In this research, we study the results about coconvex multi approximation. Recently this subject received a lot of attention. The first result in the topic is a direct theorem for the error bound of the convex approximation for functions of two continuous derivatives that were introduced by Kopotun. In 2006 Kopotun, Leviatan and Shevchuk gave definitive answers to open problems of Jackson's estimates by the moduli of smoothness of Ditzian- Totiktype, also in 2019 they stated many open problems in the coconvex case and mentioned recent developments on this subject. In convex multi approximation, we have a function f ε C[-1, 1]d change its convexity at a finite number of times in [-1, 1]d and we take care of error bound of approximation of the function f using multi algebraic polynomials, in other words, multi algebraic polynomials change its convex particularly at the same points where f is. We can get some of Jackson's estimates in which the intended constants depend on the place of the convexity change points. In order to clarify the complexities, we discuss in this research invers inequality for the coconvex approximation of the function f using multi algebraic polynomials. [ABSTRACT FROM AUTHOR]

For a nonnegative integer n, let P n be the vector space of polynomials of degree at most n, equipped with the inner product ⟨ f (x) , g (x) ⟩ = ∫ - 1 1 f (x) g (x) d x . Let B = { x n , x n - 1 , ... , 1 } , an ordered basis of P . Let { P ← n n (x) , P ← n - 1 n (x) , ⋯ , P ← 0 n (x) } be the orthogonal basis of P n obtained by applying the Gram–Schmidt procedure to B , with the resulting polynomials normalized by the condition that P ← k n (1) = 1 . We call these polynomials reverse Legendre polynomials. In this paper, we give explicit formulas for the reverse Legendre polynomials { P ← k n (x) } and determine some of their properties. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of I [ f ] = ∫ = a b f (x) d x , the Hadamard finite part of the finite-range singular integral ∫ a b f (x) d x , f (x) = g (x) / (x - t) m with a < t < b and m ∈ { 1 , 2 , ... } , assuming that (i) g ∈ C ∞ (a , b) and (ii) g(x) is allowed to have arbitrary integrable singularities at the endpoints x = a and x = b . We first prove that ∫ = a b f (x) d x is invariant under any legitimate variable transformation x = ψ (ξ) , ψ : [ α , β ] → [ a , b ] , hence there holds ∫ = α β F (ξ) d ξ = ∫ = a b f (x) d x , where F (ξ) = f (ψ (ξ)) ψ ′ (ξ) . Based on this result, we next choose ψ (ξ) such that F (ξ) , the T -periodic extension of F (ξ) , T = β - α , is sufficiently smooth, and prove, with the help of some recent extension/generalization of the Euler–Maclaurin expansion, that we can apply to ∫ = α β F (ξ) d ξ the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author: [A. Sidi, "Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions." Calcolo, 58, 2021. Article number 22]. We give a whole family of numerical quadrature formulas for ∫ = α β F (ξ) d ξ for each m, which we denote T ^ m , n (s) [ F ] . Letting G (ξ) = (ξ - τ) m F (ξ) , with τ ∈ (α , β) determined from t = ψ (τ) , and letting h = T / n , for m = 3 , for example, we have the three formulas T ^ 3 , n (0) [ F ] = h ∑ j = 1 n - 1 F (τ + j h) - π 2 3 G ′ (τ) h - 1 + 1 6 G ′ ′ ′ (τ) h , T ^ 3 , n (1) [ F ] = h ∑ j = 1 n F (τ + j h - h / 2) - π 2 G ′ (τ) h - 1 , T ^ 3 , n (2) [ F ] = 2 h ∑ j = 1 n F (τ + j h - h / 2) - h 2 ∑ j = 1 2 n F (τ + j h / 2 - h / 4). We show that all of the formulas T ^ m , n (s) [ F ] converge to I[f] as n → ∞ ; indeed, if ψ (ξ) is chosen such that F (i) (α) = F (i) (β) = 0 , i = 0 , 1 , ... , q - 1 , and F (q) (ξ) is absolutely integrable in every closed interval not containing ξ = τ , then T ^ m , n (s) [ F ] - I [ f ] = O (n - q) as n → ∞ , where q is a positive integer determined by the behavior of g(x) at x = a and x = b and also by ψ (ξ) . As such, q can be increased arbitrarily (even to q = ∞ , thus inducing spectral convergence) by choosing ψ (ξ) suitably. We provide several numerical examples involving nonperiodic integrands and confirm our theoretical results. [ABSTRACT FROM AUTHOR]