Spectral radius of the sampling operator with continuous symbol

Abstract:

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 $\mathbf {T}$. Consider the operator ${S_\varphi (m,n)}$ on $L^2(\mathbf {T})$ whose matrix with respect to the standard basis $\left \{ e^{ik\theta }:k\in \mathbf {Z} \right \}$ is given by $(a_{mi-nj})_{i,j\in \mathbf {Z}}$. 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 $\mathbf {T}$ 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.