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.

- Albert Cohen and Ingrid Daubechies,
*A stability criterion for biorthogonal wavelet bases and their related subband coding scheme*, Duke Math. J.**68**(1992), no. 2, 313–335. MR**1191564**, DOI https://doi.org/10.1215/S0012-7094-92-06814-1 - Albert Cohen and Ingrid Daubechies,
*A new technique to estimate the regularity of refinable functions*, Rev. Mat. Iberoamericana**12**(1996), no. 2, 527–591. MR**1402677**, DOI https://doi.org/10.4171/RMI/207 - Ronald G. Douglas,
*Banach algebra techniques in operator theory*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR**0361893** - T. N. T. Goodman, Charles A. Micchelli, and J. D. Ward,
*Spectral radius formulas for subdivision operators*, Recent advances in wavelet analysis, Wavelet Anal. Appl., vol. 3, Academic Press, Boston, MA, 1994, pp. 335–360. MR**1244611** - U. Grenander and G. Szegö, Toeplitz forms and their applications, Chelsea Publishing Co., New York, 1955.
- Roger A. Horn and Charles R. Johnson,
*Matrix analysis*, Cambridge University Press, Cambridge, 1985. MR**832183** - G. Strang,
*Eigenvalues of $(\downarrow \!\!2)H$ and convergence of the cascade algorithm*, IEEE Trans. Sig. Proc., 44, 1996. - W. Sweldens and P. Schröder, Building Your Own Wavelets at Home, Wavelets in Computer Graphics, ACMSIGGRAPH Course Notes, 1996.
- Lars F. Villemoes,
*Wavelet analysis of refinement equations*, SIAM J. Math. Anal.**25**(1994), no. 5, 1433–1460. MR**1289147**, DOI https://doi.org/10.1137/S0036141092228179 - P. Zizler,
*Norms of sampling operators*, Linear Algebra Appl.**277**(1998), no. 1-3, 291–298. MR**1624568**, DOI https://doi.org/10.1016/S0024-3795%2897%2910066-0

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