Generalizations of Guessab-Schmeisser formula via Fink type identity with applications to quadrature rules

TL;DR

This paper extends the Guessab-Schmeisser two-point formula for differentiable functions using Fink type identities, generalizes it for polynomial sequences, and applies these results to develop and analyze quadrature rules with error bounds.

Contribution

It introduces a generalized expansion of the Guessab-Schmeisser formula via Fink identities and applies it to derive new quadrature rules with error estimates.

Findings

Derived bounds for the generalized formulas

Developed new quadrature rules based on the expansions

Provided error bounds using Chebyshev-Gruss inequalities

Abstract

In this work, an expansion of Guessab-Schmeisser two points formula for n-times differentiable functions via Fink type identity is established. Generalization of the main result for harmonic sequence of polynomials is established. Several bounds of the presented results are proved. As applications, some quadrature rules are elaborated and discussed. Error bounds of the presented quadrature rules via Chebyshev-Gruss type inequalities are also provided.