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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Spectral radius of the sampling operator with continuous symbol
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by Mark C. Ho PDF
Proc. Amer. Math. Soc. 129 (2001), 3285-3295 Request permission

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
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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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1845004