Construction Of Quasi-Interpolation Operator By Scaling Function And Interpolatory Subdivision Schemes

Subdivision and Quasi-interpolation are important themes of approximation theory, they play a vital role in theoretical research and its applications. As we know that the core idea of Multi-resolution analysis(MRA) is approaching function gradually in different resolutions, and the double scaling function itself is an approximation subdivision scheme. It’s true that the left eigenvector of matrix M has entries h2i-j in ykM=(1/2)kyk gives the combination of scaling functions φ(x+n) that equals xk. Thus the space Vo spanned by φ(x+n) contains all polynomials of degree less than p, which provides theoretical support for building quasi-interpolation operators by scaling function. Therefore subdivision, quasi-interpolation and MRA are discussed in this paper. The main works are as follows:The first part not only presents the background and current research situation concerning subdivision and quasi-interpolation, but also focuses on the main ideas of MRA.The second part focuses on the notations and properties of one-dimensional orthogonal MRA, biorthogonal MRA, univariate B-spline and its dual functions, which provide theoretical support for constructing the mask of uniform stationary univariate interpolating subdivision scheme in part three. And the new construction of quasi-interpolation operators by the scaling function and some examples are presented.The part three mainly presents the method of constructing the mask of uniform stationary univariate interpolating subdivision scheme using the dual of scaling function. For orthogonal MRA, one method is given, which is based on the inner product between scaling function φ(x) and its translation φ(x-k)(k∈Z), especially when scaling function is B-spline, the mask based on the translates of its dual basis at half-integer. For biorthogonal MRA, the method of constructing the mask using the inner product between the biorthogonal scaling function φ(x)(the biorthogonal wavelet function yψ(x)) and the translates of its dual φ(x-k/2)(ψ(x-k/2)) is given. And some examples are presented.