紧支撑正交插值的多小波和多尺度函数

Compactly Supported Orthogonal Interpolation Multiwavelets and Multiscaling Functions

本文给出一类伸缩因子为α的紧支撑正交插值多尺度函数和多小波的构造方法.设{Vj}是尺度函数Φ(x)=[φ1(x),φ2(x),…,φa(x)]T生成的多分辨分析,Vj(?)L2(R)是{a-j/2φ(?)(ajx-k),k∈Z,(?)=1,2,…,a)线性扩张构成的子空间,其插值性是指φ1(x),φ2(x),…,φa(x)满足φj(k+(?)/a)=δk,0δj,e,j,(?)∈{1,2,…,a).当Φ(x)是正交插值的,则多分辨分析的分解或重构系数能用采样点表示而不需要用计算内积的方法产生.基于此,我们建立多小波采样定理,即如果一个连续信号f(x)∈VN,则f(x)=∑i=0a-1∑k∈Zf(k/aN+i/aN+1)φi+1(aNx-k),并给出对应多小波的显式构造公式.更进一步,证明了本文构造的多小波也有插值性.最后,还给出一个构造算例.

A general procedure for constructing a class of compactly supported orthogonal interpolation multiscaling functions and multiwavelets with dilation factor a are introduced. Let {Vj} be a multiresolution analysis generated by a multiscaling function Φ(x) = [φ1(x), φ2(x), …, φa(x)]T, where the subspace Vj denotes the L2 (R)-closed linear span of {a-j/2φ(?)(ajx - k), j,k ∈ Z, (?) = 1,2,…, a}. The interpolation property here means that φ1(x), φ2(x),…, φa(x) satisfy φj(k+(?)/a) = δk,0δj,(?),j, (?) ∈ {1,2,…, a}. When Φ(x) is orthogonal interpolation, the coefficients in the multiresolution representation can realized by sampling instead of inner products. Thereby, multiwavelets sampling theorem is established, i.e., if a continuous signal f(x) ∈ VN, then f(x) = Σi=0a-1 Σk∈Zf(k/aN+i/aN+1)φi+1 (aNx - k). The corresponding orthogonal multiwavelets are constructed explicitly. What is more, the multiwavelets we construct here are also interpolation. An example is also presented.