Interpolatory orthogonal multiwavelets and refinable functions

Multiwavelet bases of L₂ consist of families of functions {2^j/2ψ_i(2^jx - k)}. By allowing more than one function {ψ₁, ψ₂}, multiwavelets provide some useful applications in signal processing and nice features such as symmetry and orthogonality. The elementary structure for multiwavelets is the multiresolution analysis of multiplicity two {V_j} generated by dilating the basic subspace V_o. This subspace V_o is generated by a multiple refinable function φ = (φ₁, φ₂)^T (refinable vector of functions) satisfying a vector refinement equation φ(x) = Σ a(k)φ(2x-k). Here, each a(k) is a 2 × 2 matrix. In this paper, we investigate interpolatory orthogonal multiple refinable functions and multiwavelets. The interpolatory property here means that φ₁ and φ₂ vanish at all integers and half integers, except that φ₁(0) = 1 and φ₂(1/2) = 1. When φ is both interpolatory and orthogonal (which is impossible for scalar refinable functions), the coefficients in the multiresolution representation can be realized by sampling instead of inner products. If f(x) = Σ {c₁(k)φ₁(2^Nx - k) + c₂(k)φ₂(2^Nx - k)}, then c₁(k) = f(k/2^N) and c₂(k) = f(k/2^N+1/2^N+1) for k ∈ Z. What is more, the orthogonal multiwavelets we construct here are also interpolatory. We show that the refinement mask for an interpolatory orthogonal multiple refinable function and multiwavelets (filterbank) is reduced to a scalar CQF. The approximation order of interpolatory multiple refinable functions is described. A complete characterization of interpolatory orthogonal multiple refinable functions is given in this paper. However, interpolatory orthogonal multiple refinable functions cannot be symmetric. Examples are presented to illustrate the general theory.