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On the monotonicity of left and right Riemann sums

Ludovick Bouthat

Abstract

Riemann sums, a classical method for approximating the definite integral of a function, have been extensively studied in the past. However, their monotonic properties, while still of great importance, particularly in approximation theory and interpolation theory, remain somewhat obscure. This paper is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riemann sums. Additionally, we present a new insightful proof of a fundamental theorem related to these sums using tools from the theory of majorization. The author also delves deeper into a question posed by Borwein, almost resolving it completely.

On the monotonicity of left and right Riemann sums

Abstract

Riemann sums, a classical method for approximating the definite integral of a function, have been extensively studied in the past. However, their monotonic properties, while still of great importance, particularly in approximation theory and interpolation theory, remain somewhat obscure. This paper is dedicated to proving general theorems about the monotonicity of left and right Riemann sums, a problem first raised by Fejér in 1950. We provide a much-needed review of the literature on the problem and offer several new sufficient and necessary conditions for the monotonicity of Riemann sums. Additionally, we present a new insightful proof of a fundamental theorem related to these sums using tools from the theory of majorization. The author also delves deeper into a question posed by Borwein, almost resolving it completely.
Paper Structure (25 sections, 25 theorems, 150 equations)

This paper contains 25 sections, 25 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Background and context
  3. Survey of previous work
  4. The function fb
  5. Outline of the paper
  6. New general results
  7. A new enlightening refinement of Theorem 1.1
  8. A refinement of Theorem 1.12
  9. A characterization for small-order polynomials
  10. A convenient application of Fourier series
  11. A necessary condition
  12. The function fb
  13. Preliminary comments and main results
  14. Proof of Theorem 3.1 when b≠1
  15. The case b≤½
  16. ...and 10 more sections

Key Result

Theorem 1.1

MR41284MR137818 Let $f:[0,1]\to \mathbb{R}$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 43 more