可加细空间的局部基
Local bases for refinable spaces
摘要：
我们提供了一个新的表示的可加细的移位不变空间与一个composite支持的生成器，在功能与一个特殊的性质的齐性。特别地，这些函数包括生成器可再现的所有齐次多项式，这将这种表示与空间的精度联系起来。我们完全刻画了空间中的齐次函数类，并证明了它们可以再生生成元。因此，我们得出结论，齐次函数可以构造的向量相关联的频谱的规模矩阵（一个有限的方阵的元素从掩码的生成器）。此外，我们证明了转移算子的核与这个有限矩阵的核具有相同的维数。这个关系为生成器的整数平移的线性独立性提供了一个简单的测试。这在逼近论、小波理论和采样等方面有潜在的应用价值。
英文摘要：
We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular these functions include all the homogeneous polynomials that are reproducible by the generator, what links this representation to the accuracy of the space. We completely characterize the class of homogeneous functions in the space and show that they can reproduce the generator. As a result we conclude that the homogeneous functions can be constructed from the vectors associated to the spectrum of the scale matrix (a finite square matrix with entries from the mask of the generator). Furthermore, we prove that the kernel of the transition operator has the same dimension than the kernel of this finite matrix. This relation provides an easy test for the linear independence of the integer translates of the generator. This could be potentially useful in applications to approximation theory, wavelet theory and sampling.
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