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Non-separable bidimensional wavelet bases. (English) Zbl 0792.42021
The standard generalization of univariate wavelet series representations to the $$s$$-dimensional setting requires $$2^ s- 1$$ wavelets. In this paper, by considering matrix dilation in the 2-dimensional setting, where the determinant of the dilation matrix has value 2, only one wavelet is required. The approach in this paper is to construct a scaling function which solves a two-scale difference equation associated with an FIR filter. However, the regularity of the scaling function as well as its corresponding wavelet cannot be derived by the same approach as in the one-dimensional case. A review of the existing techniques to evaluate regularity is given, and new methods that allow estimation of smoothness of nonseparable solutions in the most general situations are introduced. Several interesting examples are also included in this paper.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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