Abstract Harmonic Analysis of Continuous Wavelet Transforms

Vector Valued Reproducing Kernel Hilbert Spaces and Universality

“…If Y = C, the operator A is of the form Av = v, w H for some w ∈ H, so that (W v)(x) = v, π x w H . This operator is well know in harmonic analysis as wavelet operator [17].…”

Reproducing Subgroups of $Sp(2,\mathbb{R})$ . Part I: Algebraic Classification

“…The continuous wavelet transform [7,11,20,21] and its many variants, such as, for example, the shearlet transform [5,6,14,17], 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 rôle 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.…”

Continuous Wavelets and Frames on Stratified Lie Groups I

“…The existence of admissible vectors in L 2 (G) for π was proved by Führ [13],( Corollary 5.28) for homogeneous groups. We recall this in the next Theorem:…”

“…The existence of admissible functions in L 2 was proved by Liu-Peng [26] for the Heisenberg group, and by Führ [13], (Corollary 5.28) for general homogeneous groups. (In contrast to those works, this article uses no representation theory whatsoever.)…”