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Norm Estimates for the Difference Between Bochner's Integral and the Convex Combination of Function's Values

P. Cerone, Y. J. Cho, S. S. Dragomir, J. K. Kim, S. S. Kim

TL;DR

The paper addresses estimating the error when approximating the Bochner integral of a vector-valued function by a convex combination of its values at prescribed nodes in a Banach space with the Radon-Nikodym property. It develops a kernel-based integral identity and derives Ostrowski-type bounds for vector-valued functions, with explicit dependence on $f'$, the auxiliary quantities $\mu_p$, and the chosen $L_p$ norms. The authors specialize the general results to two- and three-point rules, including trapezoidal and Simpson-type configurations, and discuss optimal node placements and weight choices to minimize the error. Overall, the work extends scalar Ostrowski inequalities to the vector-valued, Banach-space setting and provides practical, norm-based error estimates for quadrature of Banach-space-valued functions.

Abstract

Norm estimates are developed between the Bochner integral of a vector-valued function in Banach spaces having the Radon-Nikodym property and the convex combination of function values taken on a division of the interval [a,b].

Norm Estimates for the Difference Between Bochner's Integral and the Convex Combination of Function's Values

TL;DR

The paper addresses estimating the error when approximating the Bochner integral of a vector-valued function by a convex combination of its values at prescribed nodes in a Banach space with the Radon-Nikodym property. It develops a kernel-based integral identity and derives Ostrowski-type bounds for vector-valued functions, with explicit dependence on , the auxiliary quantities , and the chosen norms. The authors specialize the general results to two- and three-point rules, including trapezoidal and Simpson-type configurations, and discuss optimal node placements and weight choices to minimize the error. Overall, the work extends scalar Ostrowski inequalities to the vector-valued, Banach-space setting and provides practical, norm-based error estimates for quadrature of Banach-space-valued functions.

Abstract

Norm estimates are developed between the Bochner integral of a vector-valued function in Banach spaces having the Radon-Nikodym property and the convex combination of function values taken on a division of the interval [a,b].

Paper Structure

This paper contains 4 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. The Results
  3. The Case of Two Points
  4. The Case of Three Points

Key Result

Theorem 1

Let $\left( X;\left\| \cdot \right\| \right)$ be a Banach space with the Radon-Nikodym property and $f:\left[ a,b\right] \rightarrow X$ an absolutely continuous function on $\left[ a,b\right]$ with the property that $f^{\prime }\in L_{\infty }\left( \left[ a,b\right] ;X\right)$, i.e., Then we have the inequalities: for any $s\in \left[ a,b\right]$, where $\left( B\right) \int_{a}^{b}f\left( t\ri

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Remark 1
  • Proposition 1
  • Remark 2
  • ...and 2 more