Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets

This paper investigates spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Phi=(phi(1),...,phi(r))(T) be an r x 1 vector of compactly supported functions in L-2(R-s) satisfying Phi=Sigma(alphais an element ofZ s) a(alpha)Phi(M.-alpha), where M is an expansive integer matrix. The smoothness of Phi is measured by the Sobolev critical exponent lambda(Phi) :=sup {lambda:integral(R s)\(phi) over capj(xi)\(2)(1+\xi\(lambda))(2) dxi < infinity 1 <= j <= r}. Suppose M is similar to diag(sigma(1),...,sigma(s)) with vertical bar sigma(1)vertical bar = center dot center dot center dot = vertical bar sigma(s)vertical bar and suppa := {alpha is an element of Z(s) : a(alpha)not equal 0} is finite. For mu = (mu(1),...,mu(s)) is an element of N-0(s), define sigma(-mu) := sigma(1)(-)mu 1...sigma(s)(-mu s). Let A := Sigma(alphais an element ofZ s) a(alpha)/\detM\ and b(alpha) := Sigma(betais an element ofZ s) a(beta) circle times a(alpha+beta)/\det M\, alpha is an element of Z(s), where circle times denotes the (right) Kronecker product. Suppose that the highest total degree of polynomials reproduced by Phi is k-1 and spec(A) (the spectrum of A) is {eta(1),eta(2),...,eta(r)} with eta(1)=1 and eta(j)not equal1, 2less than or equal tojless than or equal tor. Set Ek := {eta(j)sigma(-mu), eta(j)sigma(-mu) : \mu\ M k,j=2,...,r} boolean OR {sigma(-mu) : \mu\ < 2k}. The main result of this paper asserts that if Phi is stable, then lambda(Phi) = -(log(vertical bar detM vertical bar) rho(k))s/2, where rho k := {vertical bar nu vertical bar nu is an element of spec (b(M alpha-beta))(alpha,beta is an element of K)\E-k}, and K is the set Z(s) boolean AND Sigma(infinity)(n=1) M-n(suppb). This result is obtained through an extensive use of linear algebra and matrix theory. Three examples are provided to illustrate the general theory.