Bounds for Peano kernels

Brass, Helmut

doi:10.1007/978-3-0348-6338-4_4

Cited by 15 publications

(5 citation statements)

References 10 publications

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“…in the univariate case, see (5) and (7). Explicit values for the constants γ r can be obtained by known bounds for the respective Peano kernels, see Brass (1993).…”

Section: Error Boundsmentioning

confidence: 99%

High dimensional integration of smooth functions over cubes

Novak

Ritter

1996

Numerische Mathematik

322

249

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We construct a new algorithm for the numerical integration of functions that are defined on a d -dimensional cube. It is based on the Clenshaw-Curtis rule for d = 1 and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and d ≥ 8. Classification (1991): 41A55, 41A63, 65D30, 65Y20 Mathematics Subject

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“…in the univariate case, see (5) and (7). Explicit values for the constants γ r can be obtained by known bounds for the respective Peano kernels, see Brass (1993).…”

Section: Error Boundsmentioning

confidence: 99%

High dimensional integration of smooth functions over cubes

Novak

Ritter

1996

Numerische Mathematik

322

249

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“…In this case, the functional R N (Ψ, p kl , t k ) is equivalent to the Peano constant. Theory of the Peano constants is a very important part of classical numerical theory (see [30]). Comparing the definitions of the Peano constant and optimal quadrature formulas, one can observe that the Peano constant theory is a special case of optimal algorithms theory.…”

Section: Consider the Integralmentioning

confidence: 99%

Approximate Methods for Calculating Singular and Hypersingular Integrals with Rapidly Oscillating Kernels

Roudnev

Boykova

2022

Axioms

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The article is devoted to the issue of construction of an optimal with respect to order passive algorithms for evaluating Cauchy and Hilbert singular and hypersingular integrals with oscillating kernels. We propose a method for estimating lower bound errors of quadrature formulas for singular and hypersingular integral evaluation. Quadrature formulas were constructed for implementation of the obtained estimates. We constructed quadrature formulas and estimated the errors for hypersingular integrals with oscillating kernels. This method is based on using similar results obtained for singular integrals.

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“…-If Q is an interpolatory quadrature rule with positive weights then Theorem 2 in [2] shows κ ≤ π/2q. -If Q is a Gaussian quadrature rule shifted to [0, 1] then Theorem 2 in [2] shows κ ≤ π/4q.…”

Section: Basic Quadrature Rulesmentioning

confidence: 99%

“…-If Q is a Gaussian quadrature rule shifted to [0, 1] then Theorem 2 in [2] shows κ ≤ π/4q. -If Q is a composite midpoint rule then [10] shows κ ≤ 1/2q.…”

Section: Basic Quadrature Rulesmentioning

confidence: 99%

Semi-discrete Galerkin approximation of the single layer equation by general splines

Ainsworth¹,

Grigorieff²,

Sloan³

1998

Numerische Mathematik

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This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space, and that the Peano constant of the rule is sufficiently small. It is then shown that the semidiscrete Galerkin approximation is well posed. For the trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the H 0 and H −1 norms. Mathematics Subject Classification (1991): 65N30Correspondence to: M. Ainsworth Numerische Mathematik Electronic Edition page 157 of Numer. Math. (1998) 79: 157-174 158 M. Ainsworth et al.

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Bounds for Peano kernels

Cited by 15 publications

References 10 publications

High dimensional integration of smooth functions over cubes

High dimensional integration of smooth functions over cubes

Approximate Methods for Calculating Singular and Hypersingular Integrals with Rapidly Oscillating Kernels

Semi-discrete Galerkin approximation of the single layer equation by general splines

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