arXiv:1109.6789v1 [math.GR] 30 Sep 2011 SIGNAL ANALYSES IN 2D, PART I. G. ALBERTI, L. BALLETTI, F. DE MARI, AND E. DE VITO Abstract. We classify the connected Lie subgroups of Sp(2, R) whose ele- ments have the triangular form (1.2). The classification is up to conjugation within the symplectic group Sp(2, R). Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets. 1. Introduction The continuous wavelet transform [6, 10, 19, 20] and its many variants, such as, for example, the shearlet transform [4, 5, 13, 16], lie in the background of a growing body of techniques, that may be collectively referred to as signal analysis, whose common feature is perhaps the decomposition of functions, primarily in L 2 (R d ), by means of superpositions of projections along selected “directions”. Symmetry and finite dimensional geometry often play a prominent rˆ ole in the way in which these directions are generated or selected, and hence, with this notion of signal analysis, topological transformation groups and their representations provide a natural setup. In particular, the restriction of the metaplectic representation of Sp(d, R) to its Lie subgroups produces a wealth of useful reproducing formulae [2, 3], all based on linear geometric actions either in the time or in the frequency domain, and is thus one of the most natural environments both for a unified approach and for the search of new strategies. In fact, the deep connections of the metaplectic representation with harmonic analysis in phase space is thoroughly investigated [9, 12], and one of the keys to its understanding is the Wigner transform. The central importance of the symplectic group has motivated both a general theory of “mock” metaplectic representations (and the abstract harmonic analysis thereof [7]), and a more applications-oriented approach, where the main focus is the actual study of these formulae in connection with the classical themes of signal analysis [11]. In this work, that consists of two parts, we introduce the class E of Lie subgroups of Sp(d, R) that we believe is the “right” class for signal analysis and we illustrate its relevance in 2D-analysis by exhibiting the full list of reproducing formulae that it yields, up to the appropriate notion of equivalence. In some sense, therefore, we obtain a complete picture, at least as far as continuous “geometric” transforms are concerned, of reasonable 2D signal analyses. In the first part (this paper) we classify the groups, modulo conjugation within Sp(2, R). In part II we address the analytic issues: by appealing to the theory developed in [7] we are able to show exactly which groups are reproducing and which are not. The full description of the associated admissible vectors is also achieved. In part II, we Date : July 11, 2013. 2010 Mathematics Subject Classification. Primary: 22E15,43A80, 42C40 . 1