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

Let (where is the -th Fourier coefficient of ) be a bounded measurable function on the unit circle **T**. Consider the operator on whose matrix with respect to the standard basis is given by . In this paper, we give upper and lower bound estimation for , the spectral radius of . Furthermore, we will show that in some cases (for example, if is continuous on **T** and ), the spectral radius of can be computed exactly in terms of roots of the norms of some finite Toeplitz matrices.

**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**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**

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