CONSTRUCTION OF THE MULTI-WAVELETS ON SOME SMOOTH PLANE CURVES VIA LENGTH-PRESERVING PROJECTION

Based on the theory of the discrete multi-wavelets in the space L2(R), the theory of the discrete multi-wavelets in the space L2(C) is presented properly in this paper, where C denotes a smooth plane curve. Firstly, the length-preserving projection is constructed, and by the length-preserving projection, the multiplicity multi-resolution analysis in the space L2(C) is defined properly and we define the dilation operator and translation operator in the space L2(C). Then, the two-scale refinement equations of multi-scaling function and multi-wavelet in the space L2(C) is deduced by using length-preserving mapping, the orthogonality is discussed, and the decomposition and reconstruction algorithm is computed. Finally, the example is given.

Convergence Rates of Cascade Algorithms with Infinitely Supported Masks

We investigate the solutions of refinement equations of the formwhere the function ϕ is in Lp(ℝs)(1 ≤ p ≤ ∞), a is an infinitely supported sequence on ℤs called a refinement mask, and M is an s × s integer matrix such that limn→1M–n = 0. Associated with the mask a and M is a linear operator Qa,M defined on Lp(ℝs) by Qa,Mϕ0 := Σα∈ℤsa(α)ϕ0(M · –α). Main results of this paper are related to the convergence rates of in Lp(ℝs) with mask a being infinitely supported. It is proved that under some appropriate conditions on the initial function ϕ0, converges in Lp(ℝs) with an exponential rate.