Wavelet Approximation in Orlicz Spaces

Multivariate wavelet decompositions with matrix dilations are considered. Dual wavelet frames and so-called frame-like wavelet systems (that are not frames, but preserve some important properties of frames) are discussed. Approximation properties of such systems in the Orlicz spaces are investigated. The order of approximation (in the sense of modular convergence) of wavelet frame decompositions satisfying a number of natural conditions is found for an arbitrary Orlicz space. For the Orlicz spaces satisfying $\Delta_2$-condition, an error estimate providing approximation order of such wavelet expansions is given in terms of the Luxemburg norm. Similar results are obtained for appropriate frame-like systems under the assumption that an Orlicz space satisfies the $\Delta'$-condition.