Spectral Radii of Refinement and Subdivision Operators

The spectral radii of refinement and subdivision operators considered on the space L2 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. Introduction Let q be a positive integer not equal to one, and let a(e) ∼ ∑k∈Z ake, x ∈ R, be an essentially bounded measurable function on the unit circle T, where ak denotes the k-th Fourier coefficient of a. The function a generates two operators widely used in wavelet analysis. The refinement operator D a is an operator on L2(R) defined by D a f(x) := ∑ k∈Z akf(qx− k). The subdivision operator A a associated with the function a is an operator on L2(T) with the matrix representation with respect to the standard basis {zk : z ∈ T, k ∈ Z} A a ∼  . . . . . . . . . . . . . . . . . . . . . aq a0 a−q a−2q . . . . . . a1+q a1 a1−q a1−2q . . . . . . a2+q a2 a2−q a2−2q . . . . . . . . . . . . . . . . . . . . .  . It is well known that the operators D a and A (q) a play outstanding roles in wavelet analysis [1, 6, 7] as well as in curve and surface modelling [2]. In particular, the spectral radius ρ(A a ) ofA (q) a is, in a sense, responsible for the regularity of wavelets and refinable functions [6, 8, 15]. This observation led to intensive studies of the spectral radius of A a . The most popular approach in these investigations is based on using different characteristics of finite matrices, for example, eigenvalues, norms, Received by the editors September 16, 2002 and, in revised form, November 25, 2003 and December 10, 2003. 2000 Mathematics Subject Classification. Primary 42C40, 47B35, 47B33.