A Certain Class of Weighted Approximations for Integrable Functions and Applications

Abstract

A certain class of weighted approximations, which extends the results of Masjed-Jamei [6] is introduced for integrable functions and some of upper bounds are obtained for the absolute value of the errors of such approximations in two L¹[a, b] and L^∞[a, b] spaces. As the main motivation for introducing the aforesaid class, it is shown that many new inequalities can be generated from the given error bounds. Some illustrative examples are presented in this sense. Moreover, by using the obtained error bounds, a nonstandard type of three-point weighted quadrature rules is introduced and its error bounds are computed.

Keywords:

• Error bounds
• Kernel functions
• Normed spaces
• Ostrowski and Gruss type inequalities
• Quadrature rules
• Weighted approximations

Mathematics Subject Classification:

• 26D10
• 26D20
• 41A35
• 65D30

ACKNOWLEDGMENTS

This research has been supported by a grant from “Iran National Science Foundation.”