Truncation and duality results for Hopf image algebras

Associated to an Hadamard matrix $H\in M_N(\mathbb C)$ is the spectral measure $\mu\in\mathcal P[0,N]$ of the corresponding Hopf image algebra, $A=C(G)$ with $G\subset S_N^+$. We study here a certain family of discrete measures $\mu^r\in\mathcal P[0,N]$, coming from the idempotent state theory of $G$, which converge in Ces\`aro limit to $\mu$. Our main result is a duality formula of type $\int_0^N(x/N)^pd\mu^r(x)=\int_0^N(x/N)^rd\nu^p(x)$, where $\mu^r,\nu^r$ are the truncations of the spectral measures $\mu,\nu$ associated to $H,H^t$. We prove as well, using these truncations $\mu^r,\nu^r$, that for any deformed Fourier matrix $H=F_M\otimes_QF_N$ we have $\mu=\nu$.