Raising approximation order of refinable vector by increasing multiplicity

Abstract

An algorithm is presented for raising an approximation order of any given orthogonal multiscaling function with the dilation factor α. Let Φ(x)=[φ ₁(x)Φ ₂(x),...Φr(x)]^T be an orthogonal multiscaling function with the dilation factor α and the approximation order m. We can construct a new orthogonal multiscaling function Φ ^new(x)=[Φ^T(x),Φ _r+1(x),Φ _r+2(x),...Φ _r+s (x)]^T with the approximation order m+L(L∈ℤ₊). In other words, we raise the approximation order of multiscaling function Φ(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function Φ(x)=[φ ₁(x)Φ ₂(x),...Φr(x)]^T is symmetric, then the new orthogonal multiscaling function Φ^new (x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given.