Rong-Qing Jia

Approximation with Scaled Shift-invariant Spaces by means of Quasi-projection Operators

The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this talk we we will review quasi-interpolation schemes in the setting of shift-invariant spaces. Then we will show how quasi-interpolation schemes can be developed into approximation schemes induced by quasi-projection operators. In particular, we will study approximation by quasi-projection operators in Lipschitz spaces. The study of quasi-projection operators is applied to wavelet analysis. We will discuss connections of quasi-projection operators with recent developments in wavelet analysis, such as convergence of subdivision schemes, smoothness analysis of wavelets, and the wavelet method in numerical solutions of differential equations.