On density order of quasi-projection operators and dual wavelet frames

Abstract

From two functions in the Hilbert space L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document} whose Fourier transform has certain decay, one defines a quasi-projection operator. In this paper, we prove necessary and sufficient conditions on those functions in order to the associated quasi-projection operator provides a desired approximation order and density order. To give our conditions we will use the classical notion of approximate continuity. As a consequence, we obtain approximation properties of dual wavelet frame constructed by Mixed Oblique Extension Principle. We show our results in the context of reducing subspaces of L2(Rd)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\mathbb {R}^d)$$\\end{document} with a dilation given by an expansive linear map preserving the integer lattice.