A rough estimate for the spectral radius of the sampling operator

Let ∑−∞∞ak eikθ be a bounded measurable function on the unit circle . Given , the sampling operator S (m,n) is a bounded linear operator on whose matrix with respect to the standard basis is given by . In [Proc. AMS 129 (11) (2001) 3285], a formula for the L2 spectral radius r of S (m,n) is obtained in terms of the asymptotic behavior of the supnorms of some continuous functions of when is continuous and positive on , where m=pt, n=qt, t=g.c.d.(m,n). In this paper, we shall establish an inequality that provide upper and lower bounds for r in terms of the parameter m, n and a positive eigenvalue of with maximal module, where is the restriction of S (m,n) on , the space of continuous functions on . We will also compute the actual value of r in some nontrivial cases.