Pseudo-splines, wavelets and framelets

Abstract

This thesis gives a regularity analysis of pseudo-splines of both types. In particular, an optimal estimate of the decay of the Fourier transform of the pseudo-splines is given. The regularity of pseudo-splines can then be deduced and hence, the regularity ofthe corresponding wavelets and framelets that are constructed in this thesis. We shall also show that the shifts of a pseudo-spline are linearly independent. Further, from a given pseudo-spline, a short support Riesz waveletwill be constructed, which turns out to be one of the tight framelets constructed from the same pseudo-spline by a method provided both in this thesis and one of the references. We also introduce an implementable scheme to construct smooth compactly supported biorthogonal Riesz wavelets, which satisfy any prescribed regularities, using pseudo-splines.