Diffusive wavelets on groups and homogeneous spaces

The aim of this exposition is to explain basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$. After developing a general concept for compact groups and their homogeneous spaces we give concrete examples for tori -which reflect the situation on $R^n$- and for spheres $S^2$ and $S^3$.Comment: 20 pages, 3 figure

The Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform

This note reviews complex and real techniques in harmonic analysis. We describe a common source of both approaches rooted in the covariant transform generated by the affine group. Keywords: wavelet, coherent state, covariant transform, reconstruction formula, the affine group, ax+b-group, square integrable representations, admissible vectors, Hardy space, fiducial operator, approximation of the identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal function, vertical maximal functions, non-tangential maximal functions, intertwining operator, Cauchy-Riemann operator, Laplace operator, singular integral operator, SIO, boundary behaviour, Carleson measure.Comment: 31 pages, AMS-LaTeX, no figures; v2: a major revision, sections on representations of the ax+b group and transported norms are added; v3: major revision: an outline section on complex and real variables techniques are added, numerous smaller improvements; v4: minor correction

Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization

We establish wavelet characterizations of homogeneous Besov spaces on stratified Lie groups, both in terms of continuous and discrete wavelet systems. We first introduce a notion of homogeneous Besov space $\dot{B}_{p,q}^s$ in terms of a Littlewood-Paley-type decomposition, in analogy to the well-known characterization of the Euclidean case. Such decompositions can be defined via the spectral measure of a suitably chosen sub-Laplacian. We prove that the scale of Besov spaces is independent of the precise choice of Littlewood-Paley decomposition. In particular, different sub-Laplacians yield the same Besov spaces. We then turn to wavelet characterizations, first via continuous wavelet transforms (which can be viewed as continuous-scale Littlewood-Paley decompositions), then via discretely indexed systems. We prove the existence of wavelet frames and associated atomic decomposition formulas for all homogeneous Besov spaces ${\dot B}_{p,q}^{s}$, with $1 \le p,q < \infty$ and $s \in \mathbb{R}$.Comment: 39 pages. This paper is to appear in Journal of Function Spaces and Applications. arXiv admin note: substantial text overlap with arXiv:1008.451

Extended MacMahon-Schwinger's Master Theorem and Conformal Wavelets in Complex Minkowski Space

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped one-to-one onto the future tube domain C^4_+ of the complex Minkowski space through a Cayley transformation, where other kind of (electromagnetic) wavelets have already been proposed in the literature. We study the unitary irreducible representations of the conformal group on the Hilbert spaces L^2_h(D_4,d\nu_\lambda) and L^2_h(C^4_+,d\tilde\nu_\lambda) of square integrable holomorphic functions with scale dimension \lambda and continuous mass spectrum, prove the isomorphism (equivariance) between both Hilbert spaces, admissibility and tight-frame conditions, provide reconstruction formulas and orthonormal basis of homogeneous polynomials and discuss symmetry properties and the Euclidean limit of the proposed conformal wavelets. For that purpose, we firstly state and prove a \lambda-extension of Schwinger's Master Theorem (SMT), which turns out to be a useful mathematical tool for us, particularly as a generating function for the unitary-representation functions of the conformal group and for the derivation of the reproducing (Bergman) kernel of L^2_h(D_4,d\nu_\lambda). SMT is related to MacMahon's Master Theorem (MMT) and an extension of both in terms of Louck's SU(N) solid harmonics is also provided for completeness. Convergence conditions are also studied.Comment: LaTeX, 40 pages, three new Sections and six new references added. To appear in ACH

Multidimensional Tauberian theorems for vector-valued distributions

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M-phi(f)(x, y) = (f * phi(y))(x), (x, y) is an element of R-n x R+, with kernel phi(y) (t) = y(-n)phi(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x(0)} x R-m. In addition, we present a new proof of Littlewood's Tauberian theorem