The critical point and the $p$-norm of $A_s$ and $C$-matrices

TL;DR

This paper refines bounds on the critical point for the 2-norm of $A_s$ matrices and extends the property to $p$-norms, also computing the 2-norm for certain $C$-matrices with lacunary sequences.

Contribution

It improves the bounds on the critical point $s_0$ for the 2-norm of $A_s$ matrices and demonstrates the same property for $p$-norms, also calculating the 2-norm for specific $C$-matrices.

Findings

Refined bounds for the critical point $s_0$ in the interval (1/4, 1/2).

Extended the constant 2-norm property to $p$-norms of $A_s$.

Computed the 2-norm of $C$-matrices with lacunary sequences.

Abstract

The $L$-matrix $A_s=[1/(n+s)]$ was introduced in \cite{MRtmp}. As a surprising property, we showed that its 2-norm is constant for $s \geq s_0$, where the critical point $s_0$ is unknown but relies in the interval $(1/4,1/2)$. In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding $s_0$. Moreover, we show that the same property persists for the $p$-norm of $A_s$ matrices. We also obtain the 2-norm of a family of $C$-matrices with lacunary sequences.