A Class Of Hermite Spline Wavelets

Many achievements have been made in solving integral and differential equations with wavelet bases.R.Q.Jia and S.T.Liu construct a Hermite spline wavelets fur the Sobolev space H₀¹(0,1),which are used to solve the Sturm-Liouville equation with the Dirichlet boundary condition.Motivated by Jia and Liu's work, we construct a pair of wavelets with compact supports on[-1,1],by using Hermite cubic splines.One of our wavelets is symmetric,and the other is antisymmetric. Similar methods are used to construct a wavelet bases for the Sobolev space H₀¹(0,1),with simple construction of boundary wavelets.We try to discuss the solution of a differential equation of the fourth order with boundary condition. by using our wavelets.The last section gives two negative results,which explains partly the idea of the wavelet constructions in the second part of this paper.