The p-norm of circulant matrices JMM 2022Ludovick Bouthat (Université Laval)The p-norm of circulant matrices JMM 2022 4 / 26Permutation-invariant matrix normfor every permutation matrix P and Q. ExampleEvery matrix norm induced by a permutation-invariant vector norm is a permutation-invariant matrix norm.Ludovick Bouthat (Université Laval)The p-norm of circulant matrices JMM 2022 6 / 26

Evaluating the norm of infinite matrices, as operators acting on the sequence space 2 , is not an easy task. For a few celebrated matrices, e.g., the Hilbert matrix and the Cesàro matrix, the precise value of the norm is known. But, for many other important cases we use estimated values of norm. In this note, we study the norm of L -matrices A = [a n ] , which appear in studying Hadamard multipliers of function spaces. We provide some necessary and sufficient conditions for the finiteness of norm and study the sharpness of these conditions. In particular, for the decay rate a n = O(1/n α ) , our characterization is complete. Finally, parallel to the above classical results of Hilbert and Cesàro, we succeed to show that A s = 4 for the family of L -matrices A s = [1/(n + s)] , irrelevant of the parameter s which runs over [1/2,∞) .

We show that an L -matrices A = [a n ] , with lacunary coefficients (a n ) is a bounded operator on 2 , provided that (a n ) satisfy an explicit decay rate. Moreover, by a concrete example, we see that the decay restriction is optimal. The extension to operators on p spaces, for p > 1 , is also discussed.

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In an otherwise instructive 2012 article, Szilárd András 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 (πx) are monotonically increasing relative to n, regardless of the value of p ∈ (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,