Abstract
In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if φ is an invertibleL ∞ function on the unit circle, then almost every point in σ(A *φ ) is not an eigenvalue ofA *φ . More specifically, we will show that the point spectrum ofA *φ is contained in a circle with positive radius.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Cohen and I. Daubechies,A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math. J., 1992, pp. 313–335.
A. Cohen and I. Daubechies,A new technique to estimate the regularity of refinable functions, preprint.
J.B. Conway,A Course in Functional Analysis, 2nd edition, Springer-Verlag, New York, 1990.
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.
M.C. Ho,Properties of slant Toeplitz operators, Indiana University Mathematics Journal, 45, 1996.
M. C. Ho,Spectra of slant Toeplitz operators with continuous symbol, Michigan Mathematical Journal, 44, 1997, pp. 157–166.
M. C. Ho, Ph.D. Thesis, Purdue University, 1996.
Y. D. Latushkin,On integro-functional operators with a shift which is not one-to-one, Math. USSR Izvestija, 19, 1982, pp479–493.
G. Strang,Eigenvalues of (↓2) H and convergence of the cascade algorithm, IEEE Trans. Sig. Proc., 1996.
L. Villemoes,Wavelet analysis of refinement equations, SIAM J. Maths. Analysis, 25, no. 5, 1994, pp. 1433–1460.