Construction of Orthogonal Multiscaling Functions and Multiwavelets Based on the Matrix Extension Algorithm

Yang Shouzhi, Lou Zengjian Department of Mathematics, Shantou University, Shantou 515063, P.R.China Abstract A algorithm is presented for constructing orthogonal multiscaling functions and multiwavelets with multiplicity r+s(s ≥ 1, s ∈ Z) in terms of any given orthogonal multiscaling functions with multiplicity r. That is, let Φ(x) = [φ1(x), φ2(x), · · · , φr(x)] be an orthogonal multiscaling functions with multiplicity r, with two-scale matrix symbol P (z). Then a new orthogonal multiscaling functions Φ(x) = [Φ (x), φr+1(x), φr+2(x), · · · , φr+s(x)] is constructed by applying the matrix extension method. Similarly, suppose that Ψ(x) = [ψ1(x), ψ2(x), · · · , ψr(x)] is an orthogonal multiwavelets corresponding to Φ(x), then explicit formula for constructing orthogonal multiwavelets associated with Φ(x) is obtained. In particular, the spacial case of r = s is discussed. Finally, we give some examples illustrating how to use our method to construct orthogonal multiscaling functions and the corresponding multiwavelets.