Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Spectral radius of the sampling operator with continuous symbol


Author: Mark C. Ho
Journal: Proc. Amer. Math. Soc. 129 (2001), 3285-3295
MSC (1991): Primary 42C15, 47C35, 47C38
DOI: https://doi.org/10.1090/S0002-9939-01-06057-9
Published electronically: April 9, 2001
MathSciNet review: 1845004
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42C15, 47C35, 47C38

Retrieve articles in all journals with MSC (1991): 42C15, 47C35, 47C38


Additional Information

Mark C. Ho
Affiliation: Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan
Email: hom@math.nsysu.edu.tw

Received by editor(s): December 8, 1999
Received by editor(s) in revised form: March 12, 2000
Published electronically: April 9, 2001
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society