What is the Lp norm of the $N$-dimensional Hadamard matrix $H = ((-1)^{i \cdot j})\_{i,j}$ for $p > 2$? I know that $\|H\|\_1 = N$, $\|H\|\_2 = \sqrt{N}$, $\|H\|\_\infty = N$ but I can't figure out what it is for other values of $p$. Can we at least give a good upper-bound on it?


Here I consider the induced norm: $\|H\|\_p = \max\_{x : \|x\|\_p = 1} \|Hx\|\_p$


(a previous version of this question incorrectly said that $\|H\|\_\infty = 1$)



Lp norm of Hadamard matrix
**Important Edit**: As J.J Green pointed out, the OP contains an incorrectly stated value for $\|H\|\_{\infty}$, which I copied without checking below. Interpolating between $(1,\infty)$ using the corrected version would give the trivial bound $\|H\|\_p \leq N$. You regain the sharp bound by interpolating instead between $(1,2)$ and $(2,\infty)$.




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I assume $L^p$ norm means the operator norm on $\mathbb{R}^N$ with the $\ell\_p$ norm.


Then by Riesz-Thorin-Stein interpolation, you have
$$ \|H\|\_p \leq \|H\|\_1^{1/p} \|H\|\_\infty^{1-1/p} = \sqrt[p]{N} $$
By testing on the vector $(1,0,0,\ldots,0) \mapsto (-1,1,-1,1,\ldots)$ you have
$$ \|H\|\_p \geq \sqrt[p]{N} $$


and hence $\sqrt[p]{N}$ is the value.


