Spectral radius for sampling operator
by
Mark C. Ho
Dept. of Applied Mathematics, National Sun Yat-Sen University, Taiwan

Let j(\theta) ~ \sum_\infty^\infty a_ke^ik\theta (where a_k is the k-th Fourier coefficient of j) be a bounded measurable function on the unit circle T = {e^i\theta : 0 <= \theta < 2\pi}. Let m, n be natural numbers and consider the operator S_j(m, n) on L²(T) whose matrix with respect to the standard basis {e^ik\theta : k in Z} is given by (a_mi-nj)_{i, j in Z}. The computation of the sampling operator S_j(2, 1) is essential to determining the smoothness of dyadic wavelet basis, as it has been shown by wavelet analysts. In this talk, we present the general formula for both norm of S_j(m, n)^k for all k and the spectral radius for S_j(m, n) in terms of the supnorm of the functions S_|\psi|²(p, q)^k(1) when j is continuous on T, where m = pr, n = qr, r = g.c.d.(m, n) and \psi is the average of j with respect to the mapping \tau_r(e^i\theta) = e^ir\theta. We will also provide concrete way to obtain precise value for the spectral radius when j is a trigonometric polynomial, since this is the case most frequently encountered in the application of wavelet basis.

Date received: February 24, 1999