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Abstract and classical Harmonic Analysis
Fourier analysis (group and classical - finite and infinite)
Represesntation theory (Lie groups)
Functional analysis: spectral theory
Wavelet and frame theory: Riesz bases and shift bases
Approximation and sampling theory in various contexts, for example, manifolds, Lie groups, symmetry spaces, the Heisenberg group
Shannon theory on the Heisenberg group
Study of interpolation and function spaces (spaces of Besov functions, coorbit distributions, Paley-Wiener and band-limited functions, translation-invariant spaces)
Fourier exponential bases
Interdisciplinary: astrophysics (study of CMB radiations); electronical engineering (multiplexing and demultiplexing theory)
Problems that I am currently working on them and relation between them are:
- Exponential frames and Riesz bases
- Tiling sets in finite fileds
- Paley-Wiener spaces and interpolation theory
- Wavelets and their applications in study of functions spaces on various model sets, i.e., Lie groups, manifolds, and sphere.
- Application of wavelets on the sphere in study of uncorrelation theory in statistical problems which arise in astrophysics
- Group theoretic Fourier transform and application in shift-invariant subspaces
- Sampling and interpolation theory on the Heisenberg group
- Frames and multiplexing theory
Research statement: there are two underlying themes in my research: (1) I like to construct wavelets and wavelet frames using different approaches, (2) I like to apply these wavelets and frames for various problems motivated, in particular, by experimental computer calculation, real-world physics, or both. Before starting with my research contributions, let me mention that I have a wide variety of interests and that I am often open to new collaborations and new problems.
My research interests and experience lie in harmonic analysis, focusing on representation theory, wavelet and wavelet frame theory. In general, I use functional analysis, representation theory, PDE, and spectral theory techniques for construction of "nice" wavelets with different mathematical features in various settings, and study their discretization into wavelet frames. Then I apply them for different problems, for example, for characterization of space of functions (or distributions) in various settings, sampling theory, approximation problems, and astrophysics problems on the sphere. The settings that I am interested include but are not limited to the Heisenberg group, stratified Lie groups, manifolds, symmetric spaces, and the sphere, and the class of spaces that I am interested in include interpolation spaces (Besov and Triebel-Lizorkin spaces), Hölder spaces, and coorbit spaces. In what follows I shall briefly explain some of my results.
In my PhD thesis and a series of papers, I study construction of continuous wavelets with important mathematical features and their discretization into frames in various settings that I shall explain as following. I my two papers, I focus on the Heisenberg group and develop the examples of Shannon wavelet and Mexican hat wavelet for this group using multiresolution analysis approach and spectral theory techniques, respectively. I show that the Shannon wavelet generates a Parseval frame for all of L
of the Heisenberg group, whereas the Mexican hat wavelet generates a "nice" wavelet frame. Developing a sampling theory for the Heisenberg group using the Shannon wavelet and applying Mexican hat wavelet for several problems are my ongoing projects.
In collaboration with D. Geller, in a series of papers, we focus on the construction of smooth wavelets on stratified Lie groups with compact support and the construction of Schwartz wavelets on smooth compact manifolds, specially the sphere. In these settings we also study the discretization of the wavelets into "nice" frames. A very important mathematical feature for a wavelet is to have high moments vanishing. We show that our wavelets can be chosen to have small support at high frequencies and have numerous vanishing moments in these situations, that is, the group and manifold. (We develop the notion of moment vanishing for the compact manifold). In my first joint paper with Geller, I also study Hölder spaces on the stratified Lie groups in terms of our wavelets frames. As I shall describe below, our results on the sphere have exciting application to the study of Cosmic Microwave Background (CMB) radiation.
In collaboration with J. Christensen, H. Führ, D. Geller, G. Ólafsson, and I. Pesenson, we apply our wavelets and study function and interpolation spaces on various settings (stratified Lie groups, smooth compact manifolds, symmetric spaces, abstract Hilbert spaces) in terms of smooth and band-limited wavelets. Based on our previous results, we apply the wavelet frames on stratified Lie groups and compact manifolds for characterization of Hölder spaces and Besov spaces on both settings. In collaboration with Christensen and Ólafsson, we show that the homogeneous Besov spaces on stratified Lie groups are general coorbit spaces that were initiated by Christensen and Ólafsson. An application of this result is construction of frames for the homogeneous Besov spaces and their atomic decompositions. In collaboration with Pesenson, we develop notions of bandlimitedness and smoothness of elements in an abstract Hilbert spaces and show that there is a correlation between frequency content of a function and its smoothness.
Geller and I extend the definition of Mexican hat wavelet for stratified Lie groups and compact manifolds motivated by practical applications of this wavelet on the real line. On the sphere, we call this function Mexican needlet due to the needle shape of its graph. We also generalize the definition on the sphere to generalized Mexican needlet and study their mathematical features. We show that these Mexican needlets are especially well-localized both in space and in frequency. This property results some statistical properties of these needlets and their applications which I shall review them:
In my paper on the Mexican needlets, I study some properties of generalized Mexican needlets and show that for physically reasonable CMB random fields on the sphere the Mexican needlet coefficients are asymptotically uncorrelated. This property is very important in analyzing CMB radiation data in presence of the "sky cut", the region covered by the direct radiation from the Milky Way. In my physics paper, in collaboration, I study practical applications of Mexican needlets for CMB radiation analysis in more detail and also compare them with other wavelets on the sphere, including the needlets of Narchowich and et. al. , over which they numerically appear to have superior asymptotic uncorrelation properties. Mexican needlets have already gained statisticians' and astrophysicists' attention in Europe. This news pleases us very much.
The CMB radiation has both a temperature and a polarization; the former is a scalar quantity, while the latter is a spin quantity, a section of a particular line bundle. For this reason, for analyzing the polarization part of the radiation, Geller and Marinucci generalize the construction of wavelet on the sphere to situations where spin functions (sections of line-bundles) replace ordinary scalar-valued functions. Based on the results of Geller and Marinucci, I and Geller discretize spin wavelets into nearly tight spin frames. We also show that one can choose the spin wavelets such that the spin frame element at scale a
is supported in a geodesic ball of radius Ca
. Study statistical properties of spin wavelets and their applications to CMB radiation analysis is my other ongoing project.
My special interest also includes study of wavelet theory for the one-dimensional Heisenberg group, that is, the group of 3x3 upper triangular matrices with all diagonal entries 1. This group arises in the description of one-dimensional quantum mechanical systems and has gained the attention of many mathematicians due to its analytic features and simplicity of its irreducible representations. This has motivated me to specify my study of wavelets on this group. The study resulted in a series of papers, in collaboration with B. Currey that I shall describe them briefly:
We answer an open question and initiate a notion of Heisenberg wavelet sets in the dual of the Heisenberg group by means of translation and dilation congruency. Then based on the results we discuss the sampling problem for left-invariant multiplicity free subspaces of L
of the Heisenberg group, where the group Fourier transform of each element can be identified with a function defined on C
I also study dilation properties of frames in collaboration with B. Currey. This work is motivated by the works of D. Han, D. Larson, and D. Dutkay. Han and Larson prove that for a given general Parseval frame for a Hilbert space K, there is always an orthonormal basis for a Hilbert space H with K ≤ H such that the Parseval frame can re-obtained from the projection of the orthonormal basis onto K. Based on these results, we show that any given Parseval wavelet frame in various Hilbert spaces is the projection of an orthonormal wavelet basis for a representation of generalized wavelet groups. A well-known example of these groups is the Baumslag-Solitar group whose dilation property was studied by Dutkay. In our paper, we study the dilation property for generalized wavelet groups including the case that the Heisenberg group, as translation group, is a subgroup of the wavelet groups. By this generalization, Dutkay's results becomes a special case in our situation.
Characterization of shift invariant subspaces in Hilbert spaces of functions on commutative locally compact groups has been considered by many authors since the early 90's and later. Shift invariant spaces have important applications in approximation of functions. In collaboration with B. Currey, I study the concept of shift-invariant subspace in L
for non-commutative two step nilpotent Lie groups, including the Heisenberg group, using group Fourier transformation approach. To our best knowledge, our work is original for non-commutative Lie groups and it opens many other interesting questions for our further studies.
Jens Christensen (Tufts University)
Bradley Currey (St. Louis University)
Hartmut Führ (University of Aache, RWTH Aachen, Germany)
Daryl Geller (Stony Brook University - deceased 2011)
Joshua MacArthur (Dalhousie University)
Gestur Ólafsson (Louisiana State University)
Vignon Oussa (St. Louis University)
Isaac Pesenson (University of Temple)
Mohammad Razani (City College of Technology, CUNY)
Keith Taylor (Dalhousie University)
Mathematicians with the similar research interest:
Brody Johnson (St. Louis University)
Dorin Dutkay (University of Central Florida)
Jeff Hogan (University of Newcastle, Austra)