Special B-spline Tight Framelet and It's Applications

We present a special B-spline tight frame and use it to introduce our numerical approximation method. We apply our method to investigate Gibbs effects and illustrate some features of the associated framelet expansion. It is shown that Gibbs effects occurs in the framelet expansion of a function with a jump discontinuity at 0 for certain classes of framelets. Numerical results are obtained regarding the behavior of the Gibbs effects. We present the results by expanding functions using the quasi-ane system. This system is generated by the B-spline tight framelets with a specific number of generators. We show numerically the existence of Gibbs effects in the truncated expansion of a given function by using some tight framelet representation.