Monotonicity of certain left and right Riemann sums [r-libre/2684]

Bouthat, Ludovick; Mashreghi, Javad, & Morneau-Guérin, Frédéric (2023). Monotonicity of certain left and right Riemann sums. In Alpay, Daniel; Behrndt, Jussi; Colombo, Fabrizio; Sabadini, Irene, & Struppa, Daniele C. (Ed.), Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis. IWOTA 2021, Chapman University. Birkhäuser/Springer, Cham, coll. « Operator Theory: Advances and Applications », vol. 290. ISBN 978-3-031-21459-2

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Item Type:	Book Sections
Refereed:	Yes
Status:	Published
Abstract:	In an otherwise instructive 2012 article, Szilard provided a flawed argument purportedly establishing that the left (resp. right) Riemann sum of f(x) = 1/1+x^2 with respect to the uniform partition of [0,1] into n equal intervals is monotonically decreasing (resp. increasing) relative to n. A few years later, D. Borwein, J. M. Borwein and B. Sims developed a symmetrization technique that allowed them to provide a rectified proof that the right Riemann sum of f(x) = 1/1+x^2 really is monotonically increasing relative to n. They also provided numerical evidence suggesting that the left Riemann sum is decreasing but they did not succeed in proving it. In the first part of this paper, we exploit the symmetrization technique to provide a proof that the left Riemann sum is indeed decreasing with respect to n. Subsequently, we show, using elementary calculus techniques, some trigonometry computations as well as calculations involving generalized binomial coefficients, that the left and right Riemann sums with respect to the uniform partition of [0,1] of the family of functions of the form sin^p(pi x) are monotonically increasing relative to n, regardless of the value of p in (0,2). In so doing, we answer a problem that came up in the context of foundational research on questions situated at the intersection of matrix theory and metric geometry,
Official URL:	https://link.springer.com/book/9783031214592
Depositor:	Morneau-Guérin, Frédéric
Owner / Manager:	Frédéric Morneau-Guérin
Deposited:	23 Jun 2022 13:31
Last Modified:	08 Nov 2022 13:52

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