In this note, we study the induced p-norm of circulant matrices A(n, ± a, b), acting as operators on the Euclidean space ${\mathbb{R}}^n$. For circulant matrices whose entries are nonnegative real numbers, in particular for A(n, a, b), we provide an explicit formula for the p-norm, 1 ≤ p ≤ ∞. The calculation for A(n, − a, b) is more complex. The 2-norm is precisely determined. As for the other values of p, two different categories of upper and lower bounds are obtained. These bounds are optimal at the end points (i.e. p = 1 and p = ∞) as well as at p = 2.

中文翻译：

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循环矩阵的 p 范数

在这篇文章中，我们研究了循环矩阵A ( n , ±  a , b ) 的诱导p范数，作为欧几里德空间上的算子${\mathbb{R}}^n$. 对于条目为非负实数的循环矩阵，特别是对于A ( n , a , b )，我们为p -范数提供了一个明确的公式，1 ≤  p  ≤ ∞。A ( n , −  a , b )的计算更为复杂。2-范数是精确确定的。至于p的其他值，获得了两个不同类别的上限和下限。这些边界在端点（即p  = 1 和p  = ∞）以及p  = 2 处是最佳的。