What distributions of points are possible for convergent sequences of interpolatory integration rules?

1993 ◽  
Vol 9 (1) ◽  
pp. 41-58 ◽  
Author(s):  
T. Bloom ◽  
D. S. Lubinsky ◽  
H. Stahl
Keyword(s):  
Integration Rules ◽  
Convergent Sequences

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

Some classes of ideal convergent sequences and generalized difference matrix operator

Filomat ◽  
2017 ◽  
Vol 31 (6) ◽  
pp. 1827-1834 ◽  
Author(s):  
S.A. Mohiuddine ◽  
B. Hazarika
Keyword(s):  
Matrix Operator ◽  
Difference Matrix ◽  
Convergent Sequences

Properties of Minimal Integration Rules. II

1971 ◽  
Vol 8 (3) ◽  
pp. 497-508 ◽  
Author(s):  
Nira Richter-Dyn
Keyword(s):  
Integration Rules

The Subspace of Almost Convergent Sequences

2021 ◽  
Vol 62 (4) ◽  
pp. 616-620
Author(s):  
R. E. Zvolinskii ◽  
E. M. Semenov
Keyword(s):  
Convergent Sequences

Spaces of Ideal Convergent Sequences of Bounded Linear Operators

2018 ◽  
Vol 39 (12) ◽  
pp. 1278-1290 ◽  
Author(s):  
Vakeel A. Khan ◽  
Kamal M. A. S. Alshlool ◽  
Sameera A. A. Abdullah
Keyword(s):  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Convergent Sequences

scholarly journals Revisited Optimal Error Bounds for Interpolatory Integration Rules

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
François Dubeau
Keyword(s):  
Error Bounds ◽  
Error Term ◽  
Quadrature Rules ◽  
Optimal Error ◽  
Integration By Parts ◽  
Integration Rules

We present a unified way to obtain optimal error bounds for general interpolatory integration rules. The method is based on the Peano form of the error term when we use Taylor’s expansion. These bounds depend on the regularity of the integrand. The method of integration by parts “backwards” to obtain bounds is also discussed. The analysis includes quadrature rules with nodes outside the interval of integration. Best error bounds for composite integration rules are also obtained. Some consequences of symmetry are discussed.

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