多小波采样定理的拓展

采样定理在数字信号通讯中发挥了十分重要的作用,因为信号通常由它的离散采样数据来恢复.Han Bin等人在J.Comput.Appl.Math.,2009,227:254-270]中构造了广义插值加细函数向量.本文研究与广义插值加细函数向量有关的采样定理的拓展问题.具体而言,对于已知的广义插值d-加细函数向量φ=(φ_1,…,φ_r)~T,即φe(m/r+k)=δ_kδ_(e-1-m),k∈Z,m=0,1,…,r-1,e=1,…,r我们将构造一组函数{φ_(r+1),…,φ_(dr)},使得φ~ロ=(φ~T,φ_(r+1),…,φ_(dr))~T也是d-加细的,而且满足φ_e(m/(dr)+k)=δ_kδ_(θ_(d,r(e)-m))k∈Z,m=0,1,…,dr-1,e=r+1,…,dr,其中θ_(d,r(e))=e-r+R_(e-1-r,d-1),R_(e-1-r,d-1)=「(e-1-r)/(d-1)」.我们建立与φ~■有关的采样定理.显然,φ的多小波子空间采样定理的适用范围得到了拓展.给出φ~■的多小波子空间采样级数的截断误差估计.

Extension of Multiwavelet Sampling Theorem

Sampling theorem plays a key role in digital signal communication and image processing since a continuous signal is usually recovered and processed by using its discrete samples.In this paper,we shall study the extension of sampling theorem related to generalized interpolating refinable function vectors introduced in Han Bin J.Comput.Appl.Math.,2009,227:254-270].Precisely,for an existing generalized interpolating d-refinable function vectorφ=（φ₁,...,φ_r）^T,i.e.,φe（m/r+ k） =δ_kδ_e-1-m,k∈Z,m = 0,l,...,r- 1,e= 1,...,r, we construct a set of functions {φ_r+1,...,φ_dr}such that φ<^■=(φ^T,φ_r+1,...,φ_dr)^T is also a continuous d-refinable function vector and satisfiesφ_e（m/（dr）+k）=δ_kδ_{θd,r（e）-m},k∈Z,m = 0,1,...,dr-1,e= r + 1,...,dr,withθ_d,r（e）=e-r+R_e-1-r,d-1 and R_e-1-r,d-1=（e-1-r）/（d-1）].The multiwavelet sampling theorem associated with φ^■ is established. Obviously,the sampling theorem in the multiwavelet subspaces ofφis extended.Moreover,we estimate the truncated error of the sampling theorem,which is desirable for application.