Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.
This paper deals with the design and analysis of adaptive wavelet method for the Stokes problem. First, the limitation of Richardson iteration is explained and the multiplied matrix M0 in the paper of Bramble and Pasciak is proved to be the simplest possible in an appropiate sense. Similar to the divergence operator, an exact application of its dual is shown; Second, based on these above observations, an adaptive wavelet algorithm for the Stokes problem is designed. Error analysis and computational complexity are given; Finally, since our algorithm is mainly to deal with an elliptic and positive definite operator equation, the last section is devoted to the Galerkin solution of an elliptic and positive definite equation. It turns out that the upper bound for error estimation may be improved.
We consider the problem uxx（x, t） = ut（x, t）, 0 ≤ x 〈 1, t ≥ 0, where the Cauchy data g（t） is given at x = 1. This is an ill-posed problem in the sense that a small disturbance on the boundary g（t） can produce a big alteration on its solution （if it exists）. We shall define a wavelet solution to obtain the well-posed approximating problem in the scaling space Vj. In the previous papers, the theoretical results concerning the error estimate are L2-norm and the solutions aren＇t stable at x = 0. However, in practice, the solution is usually required to be stable at the boundary. In this paper we shall give uniform convergence on interval x ∈ [0, 1].
Divergence-free wavelets play important roles in both partial differential equations and fluid mechanics.Many constructions of those wavelets depend usually on Hermite splines.We study several types of convergence of the related Hermite interpolatory operators in this paper.More precisely,the uniform convergence is firstly discussed in the second part;then,the third section provides the convergence in the Donoho's sense.Based on these results,the last two parts are devoted to show the convergence in some Besov spaces,which concludes the completeness of Bittner and Urban's expansions.
LIU YouMing & ZHAO JunJian Department of Applied Mathematics,Beijing University of Technology,Beijing 100124,China
In 2005, Garcia, Perez-Villala and Portal gave the regular and irregular sampling formulas in shift invariant space Vφ via a linear operator T between L^2（0, 1） and L^2（R）. In this paper, in terms of bases for L^2（0, α）, two sampling theorems for αZ-shift invariant spaces with a single generator are obtained.
Jun Jian ZHAO1,21.Department of Mathematics,Tianjin Polytechnic University,Tianjin 300160,P.R.China
