Abstract
Affine operators and Littlewood--Paley energy functions with matrix dilations are considered in this paper. Estimates and comparisons of the infimum and supremum measurements of these two operations are derived. These results are applied to the study of affine frames and wavelets. In particular, multivariate matrix-dilated wavelet families are characterized and a matrix-dilation oversampling theorem on preservation of frame-bound ratios is established.
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