Abstract
Abstract Although the integer translates of a box spline on the four-directional mesh do not constitute a Riesz basis, yet this box spline generates an orthonormal basis of the closed linear subspace of L 2 (R 2 ) that it spans. This paper is devoted to the study of the multiresolution analysis { V j }, with dilation matrix A = and its orthogonal complementary subspaces W j , generated by any four-directional box spline. We will show that the corresponding compactly supported semi-orthogonal wavelet does not satisfy the Riesz condition either, and that the orthonormalization of may not be in L 1 (R 2 ). The main result, however, is that by oversampling by Z 2 +, the compactly supported wavelet generates a frame of L 2 (R 2 ).
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