循环矩阵的p-范数的傅立叶分析

IF 0.3 Q4 MATHEMATICS

Concrete Operators Pub Date : 2021-01-01 DOI:10.1515/conop-2021-0123

K. R. Sahasranand

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引用次数: 3

摘要

摘要最近的一项工作导出了一类特殊循环矩阵A（n，A，b）∈n的诱导p-范数的表达式ℝn，对角项等于a∈nℝ 并且非对角线条目等于b≥0。我们使用傅立叶分析为其中的所有结果提供了较短的证明。关键的观察结果是循环矩阵被DFT矩阵对角化。结果包括ǁAǁ; p，1≤p≤∞的精确表达式，其中A=A（n，A，b），A≥0；对于A（n，−A，b），2本文章由计算机程序翻译，如有差异，请以英文原文为准。

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The p-norm of circulant matrices via Fourier analysis

Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 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. The results comprise an exact expression for ǁAǁp, 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.