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
We consider the parabolic equation with variable coefficients k（x）Uxx = ut, 0,x ≤1, t≥ 0, where 0 〈 α ≤ k（x） 〈 ＋∞, the solution on the boundary x = 0 is a given function g and ux（0,t） = 0. We use wavelet Galerkin method with Meyer multi-resolution analysis to obtain a wavelet approximating solution, and also get an estimate between the exact solution and the wavelet approximating solution of the problem.
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
