A class of generalized pseudo-splines

Zhitao Zhuang,Jianwei Yang

doi:10.1186/1029-242x-2014-359

Abstract

In this paper, a class of refinable functions is given by smoothening pseudo-splines in order to get divergence free and curl free wavelets. The regularity and stability of them are discussed. Based on that, the corresponding Riesz wavelets are constructed.

Highlights

1 Introduction We denote by Z and R the set of integers and real numbers, respectively
Combing Theorem and Proposition, we have the following theorem, which characterizes the regularity of a smoothed pseudo-spline
Theorem Let φ be the smoothed pseudo-spline of Type II with order r, m, φ(ξ ) ≤ C + |ξ | –r+κ, where κ where α = r – κ

Summary

Introduction

We denote by Z and R the set of integers and real numbers, respectively. Let Lp(R) stand for the classical Lebesgue space. Definition A multiresolution analysis of L (R) means a sequence of closed linear subspaces Vj of L (R) which satisfies (i) Vj ⊂ Vj+ , j ∈ Z, (ii) f (x) ∈ Vj if and only if f ( x) ∈ Vj+ , (iii) j∈Z Vj = L (R) and j∈Z Vj = { }, (iv) there exists a function φ ∈ L (R) such that {φ(x – k), k ∈ Z} forms a Riesz basis of V. ) becomes φ(ξ ) = a (ξ / )φ(ξ / ), ξ ∈ R. and the refinement of a pseudo-spline of Type II with order (m, ) is given by a (ξ ) := am, (ξ ) := cos m(ξ / ). We define the smoothed pseudo-spline by its refinement mask for Type I:.

Then we get the differential relation

Suppose that

One has