Spectral radius of the sampling operator with continuous symbol

Let $\varphi(\theta)\sim\sum_{-\infty}^\infty a_ke^{ik\theta}$ (where $a_k$ is the $k$-th Fourier coefficient of $\varphi$) be a bounded measurable function on the unit circle T. Consider the operator ${S_\varphi(m,n)}$ on $L^2({\mbox{\bf T}})$ whose matrix with respect to the standard basis $\left\{e^{ik\theta}:k\in{\mbox{\bf Z}}\right\}$ is given by $(a_{mi-nj})_{i,j\in{\mbox{\bf\scriptsize Z}}}$. In this paper, we give upper and lower bound estimation for $r(S_\varphi(m,n))$, the spectral radius of $S_\varphi(m,n)$. Furthermore, we will show that in some cases (for example, if $\varphi$ is continuous on T and $\varphi>0$), the spectral radius of $S_\varphi(m,n)$ can be computed exactly in terms of roots of the norms of some finite Toeplitz matrices.