The <i>p</i>-norm of circulant matrices

Bouthat, Ludovick; Khare, Apoorva; Mashreghi, Javad; Morneau-Guérin, Frédéric

doi:10.1080/03081087.2021.1983513

Cited by 5 publications

(10 citation statements)

References 33 publications

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“…As observed in [1], A " Apn, a, bq, a P R, b ě 0 is self-adjoint and hence }A} p " }A} q , for p and q satisfying 1 p `1 q " 1 (see [6]). Thus, it suffices to focus on either p P p1, 2s or p P r2, 8q.…”

Section: Results and Proofsmentioning

confidence: 99%

“…Circulant matrices arise in many applications ranging from wireless communications [2] to cryptography [3] to solving differential equations [4] (see [1] and the references therein for the historical context and more recent theoretical studies on circulant matrices). A circulant matrix is of the form , where a j P R, 1 ď j ď n. For a matrix A P R nˆn , we define the operator norm }A} p " sup x‰0 }Ax} p }x} p , for 1 ď p ď 8, where, for a vector y " py 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…We use this property of circulant matrices to study the p-norm of a special class of circulant matrices, Apn, a, bq P R nˆn , a, b P R, where Apn, a, bq :" The 1-norm and the infinity norm of Apn, a, bq are easily calculated to be |a|`pn´1q|b| by inspection. As observed in [1], it suffices to consider the following two cases: Apn, a, bq and Apn, ´a, bq where a, b ě 0. We obtain an exact expression for }A} p , 1 ă p ă 8, where A " Apn, a, bq and for }A} 2 where A " Apn, ´a, bq.…”

Section: Introductionmentioning

confidence: 99%

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The $p$-norm of circulant matrices via Fourier analysis

Sahasranand¹

2021

Preprint

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A recent paper [1] computed the induced p-norm of a special class of circulant matrices Apn, a, bq P R nˆn , with the diagonal entries equal to a P R and the off-diagonal entries equal to b ě 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. We obtain an exact expression for }A} p , 1 ď p ď 8, where A " Apn, a, bq, a ě 0 and for }A} 2 where A " Apn, ´a, bq, a ě 0; for the other p-norms of Apn, ´a, bq, 2 ă p ă 8, we provide upper and lower bounds.

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Section: Results and Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The $p$-norm of circulant matrices via Fourier analysis

Sahasranand¹

2021

Preprint

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“…For instance, in [5] and [6], the following lower and upper bounds for the Chebyshev radius were obtained:…”

Section: Proof Without Loss Of Generality Sincementioning

confidence: 99%

“…We know from Corollary 6.6 that the Chebyshev radius of Ω n relative to the metric space (Ω n , • ℓ p n →ℓ p n ) is given by J n − I n ℓ p n →ℓ p n . Motivated by this result, the operator norm from ℓ p n to ℓ p n of the matrices of the form αI n + βJ n was studied and partial results were presented in [6]. In particular, it was showed that,…”

mentioning

confidence: 99%

Hardy-type inequalities for ℓ^p sequences

Bouthat,

Mashreghi,

Morneau-Guérin

2023

Journal of Mathematical Inequalities

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In the first of this series of two articles, we studied some geometrical aspects of the Birkhoff polytope, the compact convex set of all n×n doubly stochastic matrices, namely the Chebyshev center, and the Chebyshev radius of the Birkhoff polytope associated with metrics induced by the operator norms from ℓ p n to ℓ p n for 1 ≤ p ≤ ∞. In the present paper, we take another look at those very questions, but for a different family of matrix norms, namely the Schatten p-norms, for 1 ≤ p < ∞. While studying these properties, the intrinsic connection to the minimal trace, which naturally appears in the assignment problem, is also established.

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Monotonicity of Certain Left and Right Riemann Sums

Bouthat¹,

Mashreghi²,

Morneau-Guérin³

2022

Operator Theory: Advances and Applications

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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,

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The p-norm of circulant matrices

Cited by 5 publications

References 33 publications

The $p$-norm of circulant matrices via Fourier analysis

The $p$-norm of circulant matrices via Fourier analysis

Hardy-type inequalities for ℓ^p sequences

Monotonicity of Certain Left and Right Riemann Sums

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