Error bounds for Gaussian quadrature rules using linear kernels

Abstract

It is well-known that the remaining term of a n-point Gaussian quadrature depends on the $2n$-order derivative of the integrand function. Discounting the fact that calculating a $2n$-order derivative requires a lot of differentiation, the main problem is that an error bound for a n-point Gaussian quadrature is only relevant for a function that is $2n$ times differentiable, a rather stringent condition. In this paper, by defining some specific linear kernels, we resolve this problem and obtain new error bounds (involving only the first derivative of the weighted integrand function) for all Gaussian weighted quadrature rules whose nodes and weights are pre-assigned over a finite interval. The advantage of using linear kernels is that their ${\mathcal{L}}^1$-norm, ${\mathcal{L}}^{\mathrm{\infty }}$-norm, maximum and minimum can easily be computed. Three illustrative examples are given in this direction.

Keywords:

• Gaussian quadratures
• error bounds
• linear kernels
• orthogonal polynomials
• weight function

2010 AMS Subject Classifications:

• Primary: 65D30
• Secondary: 65D05

Acknowledgements

The referee and handling editor deserve special thanks for careful reading and useful comments which have improved the manuscript. The second author thanks Tarbiat Modarres University and K.N.Toosi University of Technology for kind invitations which allowed to perform the revised version of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of I. Area has been partially supported by the Ministerio de Economía y Competitividad of Spain under grant [MTM2012–38794–C02–01], co-financed by the European Community fund FEDER.