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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]

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]

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.

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 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]

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]

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]

We construct a least squares approximation method for the recovery of complex-valued functions from a reproducing kernel Hilbert space on D ⊂ R d . The nodes are drawn at random for the whole class of functions, and the error is measured in L 2 (D , ϱ D) . We prove worst-case recovery guarantees by explicitly controlling all the involved constants. This leads to new preasymptotic recovery bounds with high probability for the error of hyperbolic Fourier regression on multivariate data. In addition, we further investigate its counterpart hyperbolic wavelet regression also based on least squares to recover non-periodic functions from random samples. Finally, we reconsider the analysis of a cubature method based on plain random points with optimal weights and reveal near-optimal worst-case error bounds with high probability. It turns out that this simple method can compete with the quasi-Monte Carlo methods in the literature which are based on lattices and digital nets. [ABSTRACT FROM AUTHOR]

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals I [ f ] = ∫ = a b f (x) d x , where f (x) = g (x) / (x - t) 3 , assuming that g ∈ C ∞ [ a , b ] and f(x) is T-periodic, T = b - a . With h = T / n , these numerical quadrature formulas read T ^ n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , T ^ n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , T ^ 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). We also showed that these formulas have spectral accuracy; that is, T ^ n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In the present work, we continue our study of these formulas for the special case in which f (x) = cos π (x - t) T sin 3 π (x - t) T u (x) , where u(x) is in C ∞ (R) and is T-periodic. Actually, we prove that T ^ n (s) [ f ] , s = 0 , 1 , 2 , are exact for a class of singular integrals involving T-periodic trigonometric polynomials of degree at most n - 1 ; that is, T ^ n (s) [ f ] = I [ f ] when f (x) = cos π (x - t) T sin 3 π (x - t) T ∑ m = - (n - 1) n - 1 c m exp (i 2 m π x / T). We also prove that, when u(z) is analytic in a strip | Im z | < σ of the complex z-plane, the errors in all three T ^ n (s) [ f ] are O (e - 2 n π σ / T) as n → ∞ , for all practical purposes. [ABSTRACT FROM AUTHOR]

We consider the numerical computation of finite-range singular integrals that are defined in the sense of Hadamard Finite Part, assuming that g ∈ C ∞ [ a , b ] and f (x) ∈ C ∞ (R t) is T-periodic with f ∈ C ∞ (R t) , R t = R \ { t + k T } k = - ∞ ∞ , T = b - a . Using a generalization of the Euler–Maclaurin expansion developed in [A. Sidi, Euler–Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities. Math. Comp., 81:2159–2173, 2012], we unify the treatment of these integrals. For each m, we develop a number of numerical quadrature formulas T ^ m , n (s) [ f ] of trapezoidal type for I[f]. 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 show 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. We provide a numerical example involving a periodic integrand with m = 3 that confirms our convergence theory. We also show how the formulas T ^ 3 , n (s) [ f ] can be used in an efficient manner for solving supersingular integral equations whose kernels have a (x - t) - 3 singularity. A similar approach can be applied for all m. [ABSTRACT FROM AUTHOR]

We introduce a new quadrature rule based on Chebyshev's and Simpson's rules. The corresponding composite rule induces the adaptive method of approximate integration. We propose a stopping criterion for this method and we prove that if it is satisfied for a function which is either 3-convex or 3-concave, then the integral is approximated with the prescribed tolerance. Nevertheless, we give an example of a function which does satisfy our criterion, but the approximation error exceeds the assumed tolerance. The numerical experiments (performed by a computer program created by the author) show that integration of 3-convex functions with our method requires considerably fewer steps than the adaptive Simpson's method with a classical stopping criterion. As a tool in our investigations we present a certain inequality of Hermite–Hadamard type. [ABSTRACT FROM AUTHOR]

In this paper, we use the Nyström method based on a quadrature formula associated with non-uniform quasi-interpolation for solving Hammerstein integral equations. This method results in a system of nonlinear equations for determining approximate values of the exact solution of the integral equation. Numerical examples are given for illustrating the performance of the method. [ABSTRACT FROM AUTHOR]

The Euler-Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum ∑k=0n-1f(k) of values of a function f by a linear combination of a corresponding integral of f and values of its higher-order derivatives f(j). An alternative (Alt) summation formula is proposed, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of f. Explicit and rather easy to use bounds on the remainder are given. Extensions to multi-index summation and to sums over lattice polytopes are indicated. Applications to summing possibly divergent series are presented. The Alt formula will in most cases outperform, or greatly outperform, the EM summation formula in terms of the execution time and memory use. One of the advantages of the Alt calculations is that, in contrast with the EM ones, they can be almost completely parallelized. Illustrative examples are given. In one of the examples, where an array of values of the Hurwitz generalized zeta function is computed with high accuracy, it is shown that both our implementation of the EM summation formula and, especially, the Alt formula perform much faster than the built-in Mathematica command HurwitzZeta[]. [ABSTRACT FROM AUTHOR]

A new quadrature rule for functions defined on the sphere is introduced. The nodes are defined as the points of the Cubed Sphere. The associated weights are defined in analogy to the trapezoidal rule on each panel of the Cubed Sphere. The formula enjoys a symmetry property ensuring that a proportion of 7/8 of all Spherical Harmonics is integrated exactly. Based on the remaining Spherical Harmonics, it is possible to define modified weights giving an enhanced quadrature rule. Numerical results show that the new quadrature is competitive with classical rules of the literature. This second quadrature rule is believed to be of interest for applied mathematicians, physicists and engineers dealing with data located at the points of the Cubed Sphere. [ABSTRACT FROM AUTHOR]

Modified Stieltjes polynomials are defined and used to construct suboptimal extensions of Gaussian rules with one or two degrees less of polynomial exactness than the corresponding Kronrod extension. We prove that, for wide classes of weight functions and a sufficiently large number of nodes, the extended quadratures have positive weights and simple nodes on the interval $$[-1,1]$$ . The classes of weight functions considered complement those for which the Gauss-Kronrod rule is known to exist. Also, strong asymptotic representations on the whole interval $$[-1,1]$$ are given for the modified Stieltjes polynomials, which prove that they behave asymptotically as orthogonal polynomials. Finally, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

The Filon-Clenshaw-Curtis method (FCC) for the computation of highly oscillatory integrals is known to attain surprisingly high precision. Yet, for large values of frequency $$\omega $$ it is not competitive with other versions of the Filon method, which use high derivatives at critical points and exhibit high asymptotic order. In this paper we propose to extend FCC to a new method, FCC $$+$$ , which can attain an arbitrarily high asymptotic order while preserving the advantages of FCC. Numerical experiments are provided to illustrate that FCC $$+$$ shares the advantages of both familiar Filon methods and FCC, while avoiding their disadvantages. [ABSTRACT FROM AUTHOR]

We investigate the impact of adding inner nodes for a Filon-type method for highly oscillatory quadrature. The error of Filon-type method is composed of asymptotic and interpolation errors, and the interplay between the two varies for different frequencies. We are particularly concerned with two strategies for the choice of inner nodes: Clenshaw-Curtis points and zeros of an appropriate Jacobi polynomial. Once the frequency $$\omega $$ is large, the asymptotic error dominates, but the situation is altogether different when $$\omega \ge 0$$ is small. In the first regime, our optimal error bounds indicate that Clenshaw-Curtis points are always marginally better, but this is reversed for small $$\omega $$ ; then, Jacobi points enjoy an advantage. The main tool in our analysis is the Peano Kernel theorem. While the main part of the paper addresses integrals without stationary points, we indicate how to extend this work to the case when stationary points are present. Numerical experiments are provided to illustrate theoretical analysis. [ABSTRACT FROM AUTHOR]

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ as the weight functions on [0, m] ( m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated. [ABSTRACT FROM AUTHOR]

A hypersingular integral can be regularized by replacing the whole integrand by a forward difference quotient of 2nd order. If the density function is nearly singular, then Gauss quadrature formulas associated with a suitable modification of the Chebyshev weight function allow to obtain great precision with few nodes. However, in most cases, the own nature of this procedure makes unpredictable the location of quadrature nodes. This paper presents a simple but effective technique whose aim is to mitigate instability when some node lies too close to the pole. Some numerical examples are shown to evaluate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

A class of Fredholm integral equations of the second kind, with respect to the exponential weight function $$w(x)=\exp (-(x^{-\alpha }+x^\beta ))$$ , $$\alpha >0$$ , $$\beta >1$$ , on $$(0,+\infty )$$ , is considered. The kernel k( x, y) and the function g( x) in such kind of equations, can grow exponentially with respect to their arguments, when they approach to $$0^+$$ and/or $$+\infty $$ . We propose a simple and suitable Nyström-type method for solving these equations. The study of the stability and the convergence of this numerical method in based on our results on weighted polynomial approximation and 'truncated' Gaussian rules, recently published in Mastroianni and Notarangelo (Acta Math Hung, 142:167-198, 2014), and Mastroianni, Milovanović and Notarangelo (IMA J Numer Anal 34:1654-1685, 2014) respectively. Moreover, we prove a priori error estimates and give some numerical examples. A comparison with other Nyström methods is also included. [ABSTRACT FROM AUTHOR]

A Chebyshev-type quadrature for a given weight function is a quadrature formula with equal weights. In this work, we show that a method presented by Kane may be used to determine the order of magnitude of the minimal number of nodes required in Chebyshev-type quadratures for doubling weight functions. This extends a long line of research on Chebyshev-type quadratures starting with the 1937 work of Bernstein. [ABSTRACT FROM AUTHOR]

We compute semi-explicitly or recursively the weights of the generalized Gauss-Radau and Gauss-Lobatto quadratures with Jacobi weight functions. [ABSTRACT FROM AUTHOR]

We construct several sequences of asymptotically optimal definite quadrature formulae of fourth order and evaluate their error constants. Besides the asymptotical optimality, an advantage of our quadrature formulae is the explicit form of their weights and nodes. For the remainders of our quadrature formulae monotonicity properties are established when the integrand has a non-negative or non-positive fourth derivative, and a-posteriori error estimates are proven. [ABSTRACT FROM AUTHOR]

We consider the Gauss-Radau quadrature formulae with $$c=-1$$ or $$c=1$$ , for the Bernstein-Szegő weight functions consisting of anyone of the four Chebyshev weights divided by the polynomial $$\rho (t)=1-\frac{4\gamma }{(1+\gamma )^2}\,t^2, t\in (-1,1),\ -1<\gamma \le 0$$ . For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points $$\mp 1$$ and a sum of semi-axes $$\rho >1$$ , for the given quadrature formula. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective error bounds for this quadrature formula. An alternative approach, which has initiated this research, has been proposed recently by Notaris (Math Comp , ). [ABSTRACT FROM AUTHOR]

In this paper, we present high accuracy quadrature formulas for hyper-singular integrals $\int_{a}^{b}g(x)q^{\alpha}(x,t)\, dx$, where $q(x,t)=|x-t|$ (or $x-t$), $t\in(a,b)$, and $\alpha\leq-1$ (or $\alpha<-1$). If $g(x)$ is $2m+1$ times differentiable on $[a,b]$, the asymptotic expansions of the error show that the convergence order is $O(h^{2\mu+1+\alpha})$ with $q(x,t)=|x-t|$ (or $x-t$) for $\alpha\leq-1$ (or $\alpha<-1$ and α being non-integer), and the error power is $O(h^{\eta})$ with $q(x,t)=x-t$ for α being integers less than −1, where $\eta =\min(2\mu,2\mu+2+\alpha)$ and $\mu=1,\ldots,m$. Since the derivatives of the density function $g(x)$ in the quadrature formulas can be eliminated by means of the extrapolation method, the formulas can easily be applied to solving corresponding hyper-singular boundary integral equations. The reliability and efficiency of the proposed formulas in this paper are demonstrated by some numerical examples. [ABSTRACT FROM AUTHOR]

We construct cubatures that approximate the integral of a function u over the unit ball by the linear combination of surface integrals over the unit sphere of normal derivatives of u and surface integrals of u and Δ 2 u over m spheres, centered at the origin. We derive explicitly the weights and the nodes of these cubatures, and show that they are exact for all ( 2 m + 2 ) -harmonic functions. [ABSTRACT FROM AUTHOR]

In this paper we give a survey of some results concerning the computation of quadrature formulas on the unit circle. Like nodes and weights of Gauss quadrature formulas (for the estimation of integrals with respect to measures on the real line) can be computed from the eigenvalue decomposition of the Jacobi matrix, Szegő quadrature formulas (for the approximation of integrals with respect to measures on the unit circle) can be obtained from certain unitary five-diagonal or unitary Hessenberg matrices that characterize the recurrence for an orthogonal (Laurent) polynomial basis. These quadratures are exact in a maximal space of Laurent polynomials. Orthogonal polynomials are a particular case of orthogonal rational functions with prescribed poles. More general Szegő quadrature formulas can be obtained that are exact in certain spaces of rational functions. In this context, the nodes and the weights of these rules are computed from the eigenvalue decomposition of an operator Möbius transform of the same five-diagonal or Hessenberg matrices. [ABSTRACT FROM AUTHOR]

We construct a class of cubature formulae for harmonic functions on the unit disk based on line integrals over $$2n+1$$ distinct chords. These chords are assumed to have constant distance $$t$$ to the center of the disk, and their angles to be equispaced over the interval $$[0,2\pi ]$$ . If $$t$$ is chosen properly, these formulae integrate exactly all harmonic polynomials of degree up to $$4n+1$$ , which is the highest achievable degree of precision for this class of cubature formulae. For more generally distributed chords, we introduce a class of interpolatory cubature formulae which we show to coincide with the previous formulae for the equispaced case. We give an error estimate for a particular cubature rule from this class and provide numerical examples. [ABSTRACT FROM AUTHOR]

We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness [ABSTRACT FROM AUTHOR]

A new endpoint-corrected rule for the Clenshaw-Curtis (C-C) quadrature is proposed to improve the convergence rate. The error behavior is compared, analytically and numerically, to the C-C rule and related quadrature rules: the Fejér rules of the first and second kind and the Basu rule. [ABSTRACT FROM AUTHOR]

A method for fast evaluation of band-limited functions (spherical polynomials) at many scattered points on the unit 2-d sphere is presented. The method relies on the superb localization of the father needlet kernels and their compatibility with spherical harmonics. It is fast, local, memory efficient, numerically stable and with guaranteed (prescribed) accuracy. The speed is independent of the band limit and depends logarithmically on the prescribed accuracy. The method can be also applied for approximation on the sphere, verification of spherical polynomials and for fast generation of surfaces in computer-aided geometric design. It naturally extends to higher dimensions. [ABSTRACT FROM AUTHOR]

In addition to some new cubature formulas for the approximation of integrals over the unit disk, we present a survey of all known cubature formulas of algebraic degree for this region. [ABSTRACT FROM AUTHOR]