A rough estimate for the spectral radius of the sampling operator

✍ Scribed by Mark C. Ho

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 216 KB
• Volume: 375
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00559-7

✦ Synopsis

Let ϕ ∼ ∞ -∞ a k e ikθ be a bounded measurable function on the unit circle T. Given m, n ∈ N, the sampling operator S ϕ (m, n) is a bounded linear operator on L 2 (T) whose matrix with respect to the standard basis {e k (z) = z k : k ∈ Z} is given by (a mi-nj ) i,j ∈Z . In [Proc. AMS 129 (11) (2001) 3285], a formula for the L 2 spectral radius r of S ϕ (m, n) is obtained in terms of the asymptotic behavior of the supnorms of some continuous functions of when ϕ is continuous and positive on T, 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 S ϕ (m, n)| C(T) with maximal module, where S ϕ (m, n)| C(T) is the restriction of S ϕ (m, n) on C(T), the space of continuous functions on T. We will also compute the actual value of r in some nontrivial cases.