On the monotonicity of left and right Riemann sums

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 Riema...

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Bibliographic Details
Published in:Sampling theory, signal processing, and data analysis Vol. 22; no. 1
Main Author: Bouthat, Ludovick
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.06.2024
Subjects:
Abstract Harmonic Analysis
Machine Learning
Mathematics
Mathematics and Statistics
Original Article
Signal,Image and Speech Processing
26A42
Monotone sequences
Riemann sums
26A48
Interpolation in approximation theory
41A05
26D15
ISSN:2730-5716, 2730-5724
Online Access:Get full text
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Summary: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. Lastly, we delve deeper into a question posed by Borwein, almost resolving it completely.
ISSN:2730-5716
2730-5724
DOI:10.1007/s43670-024-00088-8