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

  • 1. A. Cohen and I. Daubechies, A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math. J., 68, 1992, pp. 313-335. MR 94b:94005
  • 2. A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Revista Mathematica Iberoamericana, 12, 1996, pp. 527-591. MR 97g:42025
  • 3. R.G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972. MR 50:14335
  • 4. T. Goodman, C. Micchelli and J. Ward, Spectral radius formula for subdivision operators, Recent Advances in Wavelet Analysis, ed. L. Schumaker and G. Webb, Academic Press, 1994, pp. 335-360. MR 94m:47076
  • 5. U. Grenander and G. Szegö, Toeplitz forms and their applications, Chelsea Publishing Co., New York, 1955.
  • 6. R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University Press, 1985. MR 87e:15001
  • 7. G. Strang, Eigenvalues of $(\downarrow2)H$ and convergence of the cascade algorithm, IEEE Trans. Sig. Proc., 44, 1996.
  • 8. W. Sweldens and P. Schröder, Building Your Own Wavelets at Home, Wavelets in Computer Graphics, ACMSIGGRAPH Course Notes, 1996.
  • 9. L. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Analysis, 25, no. 5, 1994, pp. 1433-1460. MR 96f:39009
  • 10. P. Zizler, Norm of sampling operators, Linear Algebra and its Application, 277, 1998, pp. 291-298. MR 99c:47030

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

DOI: https://doi.org/10.1090/S0002-9939-01-06057-9
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

American Mathematical Society