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Gruss Type Discrete Inequalities in Normed Linear Spaces, Revisited

Sever Silvestru Dragomir

TL;DR

The paper revisits Grüss-type inequalities for sequences in real or complex normed spaces under boundedness constraints, deriving sharp constants and unifying several variants. It then applies these bounds to discrete transforms and polynomials, producing explicit error terms for the discrete Fourier transform, the discrete Mellin transform, and vector-valued polynomials. The results provide a cohesive framework with optimal constants (notably $1$ and $\tfrac{1}{2}$) and yield practical, sharp bounds for transform approximations and polynomial evaluations in numerical analysis. These contributions have potential impact on error estimation in signal processing and vector-valued numerical methods where weighted sums, transforms, and polynomial evaluations arise.

Abstract

Some sharp inequalities of Gruss type for sequences of vectors in real or complex normed linear spaces are obtained. Applications for the discrete Fourier and Mellin transform are given. Estimates for polynomials with coefficients in normed spaces are provided as well.

Gruss Type Discrete Inequalities in Normed Linear Spaces, Revisited

TL;DR

The paper revisits Grüss-type inequalities for sequences in real or complex normed spaces under boundedness constraints, deriving sharp constants and unifying several variants. It then applies these bounds to discrete transforms and polynomials, producing explicit error terms for the discrete Fourier transform, the discrete Mellin transform, and vector-valued polynomials. The results provide a cohesive framework with optimal constants (notably and ) and yield practical, sharp bounds for transform approximations and polynomial evaluations in numerical analysis. These contributions have potential impact on error estimation in signal processing and vector-valued numerical methods where weighted sums, transforms, and polynomial evaluations arise.

Abstract

Some sharp inequalities of Gruss type for sequences of vectors in real or complex normed linear spaces are obtained. Applications for the discrete Fourier and Mellin transform are given. Estimates for polynomials with coefficients in normed spaces are provided as well.

Paper Structure

This paper contains 5 sections, 17 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Some Analytic Inequalities
  3. Application for Discrete Fourier Transforms
  4. Application for the Discrete Mellin Transform
  5. Application for Polynomials

Key Result

Theorem 1

Let $\left( X,\left\Vert \cdot \right\Vert \right)$ be a normed linear space over the real or complex number field $\mathbb{K}$$\left( \mathbb{K}=\mathbb{R},\mathbb{C}\right) ,$$\overline{\mathbf{\alpha }}=\left( \alpha _{1},\dots ,\alpha _{n}\right) \in \mathbb{R}^{n},$$\overline{\mathbf{p}}=\left( The constant $1$ in the first branch, $\frac{1}{2}$ in the second branch and $1$ in the third branc

Theorems & Definitions (35)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Remark 1
  • Corollary 2
  • Theorem 3
  • proof
  • ...and 25 more