(untitled) (Journal of Computational Mathematics, 1997)
This paper gives a kind of series representation of the scaling functions φ_N and the associated wavelets ψ _N constructed by Daubechies. Based on Poission summation formula, the functions φ_N(x+N-1), φ_N(x+N), …, φ_N(x+2N-2) (0 ≤ x≤ 1) are linearly represented by φ_N(x), φ_N(x+1), …, φ_N(x+2N-2) and some polynomials of order less than N, and Φ _0(x):=(φ_N(x), φ_N(x+1), …,φ_N(x+N-2))^t is translated into a solution of a nonhomogeneous vector-valued functional equation f(x)=A_d f(2x-d)+P _d(x),x[d2,d+12], d=0,1, where A_0,A_1 are (N-1)× (N-1)-dimensional matrices, the components of P_0(x), P_1(x) are polynomials of order less than N. By iteration, Φ_0(x) is eventually represented as an (N-1)-dimensional vector series _k=0 ^∞ u _k(x) with vector norm \|U _k(x)\|≤ Cβ ^k, where β =β_N