小波标架的稳定性

On the Stability of Wavelet Frames

在小波分析理论中，标架起着十分重要的作用．对(∈Ｌ~２（Ｒ）和ａ＞１．ｂ＞０，Ｉ．Ｄａｕｂｅｃｈｉｅｓ给出了｛ａ^（ｊ／２）(（ａ~ｊｘ—ｋｂ）：ｊ，ｋ ∈Ｚ｝构成Ｌ~２（Ｒ）的标架的充分条件．近年来，人们对小波标架的稳定性进行了大量研究．首先把Ｋａｄｅｃ定理推广到高维情形，然后研究当(，｛ａ~ｊ｝，｛ｋ｝同时变化时标架的稳定性．特别地，我们给出｛ａ~ｊ｝扰动时标架的稳定性．

The theory of frames is very important for wavelet analysis. For ( ∈ L^2(R) and a > 1, b > 0, I. Daubechies gave a sufficient condition ensuring {a^(j/2) ((a^j x - kb): j, k ∈ Z } to be a frame for L^2(R). Recently, much effort has spent on the study of the stability of wavelet frames. In this paper, after obtaining a multivariate version of Kadec's 1/4- theorem, we study the stability of wavelet frames when (, {a^j} and {k} have some perturbation simultaneously. In particular, we study the effect of the perturbation to {a^j}.