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We define the vector-valued, matrix-weighted function spacesḞ αq p (W ) (homogeneous) and F αq p (W ) (inhomogeneous) on R n , for α ∈ R, 0 < p < ∞, 0 < q ≤ ∞, with the matrix weight W belonging to the A p class. For 1 < p < ∞, we show that L p (W ) =Ḟ 02 p (W ), and, for k ∈ N, that F k2 p (W ) coincides with the matrix-weighted Sobolev space L p k (W ), thereby obtaining Littlewood-Paley characterizations of L p (W ) and L p k (W ). We show that a vectorvalued function belongs toḞ αq p (W ) if and only if its wavelet or ϕ-transform coefficients belong to an associated sequence spaceḟ αq p (W ). We also characterize these spaces in terms of reducing operators associated to W .

A new theory is introduced for analyzing image compression methods that are based on compression of wavelet decompositions. This theory precisely relates a) the rate of decay in the error between the original image and the compressed image (measured in one of a family of so-called L p norms) as the size of the compressed image representation increases (i.e., as the amount of compression decreases) to b) the smoothness of the image in certain smoothness classes called Besov spaces. Within this theory, the error incurred by the quantization of wavelet transform coefficients is explained. Several compression algorithms based on piecewise constant approximations are analyzed in some detail. It is shown that if pictures can be characterized by their membership in the smoothness classes considered here, then wavelet-based methods are near optimal within a larger class of stable (in a particular mathematical sense) transform-based, nonlinear methods of image compression. Based on previous experimental research on the spatial-frequencyintensity response of the human visual system, it is argued that in most instances the error incurred in image compression should be measured in the integral ( L ' ) sense instead of the mean-square ( L~) sense. Index Terms-Image compression, wavelets, smoothness of images, quantization.

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