Construction of orthonormal multi-wavelets with additional vanishing moments

Abstract

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ ^♯:=[φ ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ ^♯ are constructed in such a way that φ ^♯ has p.p.o.=n>m and their two-scale symbols P ^♯ and Q ^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ ^D_m , then φ ^♯:=[φ ^D_m ,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ ^♯=[φ ^⊤,φ₃]^⊤ and multi-wavelet ψ ^♯ that not only increase the order of vanishing moments but also preserve symmetry.