Spectral radii of refinement and subdivision operators

Abstract

Abstract. The spectral radii of refinement and subdivision operators considered on the space L 2 can be estimated by using norms of their symbols. In several cases, including those arising in wavelet analysis, the exact value of the spectral radius is found. For example, if T is the unit circle and if the symbol a of a refinement operator satisfies the conditions |a(z)| 2 + |a(−z)| 2 = 4, z ∈ T, and a(1) = 2, then the spectral radius of this operator is equal to √ 2.

The bounds estimate of sub-band operators for multi-band wavelets

“…During the last three decades, the theory of frames, which generalize the notion of bases by allowing redundancy yet still providing a reconstruction formula, has been growing rapidly, since several new applications such as nonlinear sparse approximation (e.g., image compression), coarse quantization, data transmission with erasures, and wireless communication, have been developed [ 1 – 7 ]. As a special class of frames, the multi-band wavelets have attracted considerable attention due to their richer parameter space, to give better energy compaction than 2-band wavelets [ 8 – 16 ].…”