Regularized Solutions of a Nonlinear Convolution Equation on the Euclidean Group

In this paper we apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation. First, we review the theory of the irreducible unitary representations of the motion group and discuss the corresponding Fourier transform of functions on the motion group. The main reasons why exact solutions of this convolution equation do not exist in many cases are discussed. Techniques for regularization of the problem and numerical methods for finding approximate solutions are presented. Examples are considered and approximate solutions are found.     

The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups

In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n ⊗ R^m thereby formulate the Fourier transform on H_n ⊗ R^m (n ≥ 0, m ≥ 0}, which naturally unifies the Fourier transform between the Euclidean group and the Heisenberg group, more generally, between Abelian groups and two-step nilpotent Lie groups. Moreover, by the Plancberel formula for H_n ⊗ R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n ⊗ R^m.     

Racah operators for the group of motions

The author [5] introduced Racah operators for unitary representations of topological groups of type I. In the present paper, we indicate the explicit form of these operators for representations of the group of motions of three-dimensional Euclidean space.     

On inversion and Sobolev estimate results about the weighted Radon transform

In this paper, we derive an inversion of the weighted Radon transform by Fourier transform, Riesz potential, and integral transform. We extend results of Rigaud and Lakhal to the n‐dimensional Euclidean space. Furthermore, we obtain some properties of the weighted Radon transform. Finally, we develop some estimate results of the weighted Radon transform under Sobolev space.     

幂零Lie群上的不定性原理

设N是具有平方可积表示的幂零Lie群,是其Plancherel测度．本文将N上群Fourier变换矩阵化,并由此给出N上不定性原理的一种定量描述．此外,还对N上不定性原理的定性描述（简称QUP）作了讨论,结果显示出N上QUP与P（λ）的零点集之代数、几何性质的一些联系.     

Plancherel Inversion as Unified Approach to Wavelet Transforms and Wigner Functions

We demonstrate that the Plancherel transform for Type-I groups provides one with a natural, unified perspective for the generalized continuous wavelet transform, on the one hand, and for a class of Wigner functions, on the other. We first prove that a Plancherel inversion formula, well known for Bruhat functions on the group, holds for a much larger class of functions. This result allows us to view the wavelet transform as essentially the inverse Plancherel transform. The wavelet transform of a signal is an L²-function on an appropriately chosen group while the Wigner function is defined on a coadjoint orbit of the group and serves as an alternative characterization of the signal, which is often used in practical applications. The Plancherel transform maps L²-functions on a group unitarily to fields of Hilbert-Schmidt operators, indexed by unitary irreducible representations of the group. The wavelet transform can essentially be looked upon as a restricted inverse Plancherel transform, while Wigner functions are modified Fourier transforms of inverse Plancherel transforms, usually restricted to a subset of the unitary dual of the group. Some known results on both Wigner functions and wavelet transforms, appearing in the literature from very different perspectives, are naturally unified within our approach. Explicit computations on a number of groups illustrate the theory. Communicated by Gian Michele Graf submitted 05/06/01, accepted: 19/09/02     

Separation of variables and the computation of Fourier transforms on finite groups, I

This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroup-adapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easy-to-use upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest.

Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the best known fast transforms for some abelian groups, the symmetric groups and their wreath products, and the classical Weyl groups. Beyond this, we obtain greatly improved upper bounds for the general linear and unitary groups over a finite field, and for the classical Chevalley groups over a finite field. We also apply these techniques to obtain analogous results for homogeneous spaces.

This is part I of a two part paper. Part II will present a refinement of these techniques which yields further savings.

     

Wavelet transform and radon transform on the quaternion Heisenberg group

Let Q be the quaternion Heisenberg group,and let P be the affine automorphism group of Q.We develop the theory of continuous wavelet transform on the quaternion Heisenberg group via the unitary representations of P on L2(Q).A class of radial wavelets is constructed.The inverse wavelet transform is simplified by using radial wavelets.Then we investigate the Radon transform on Q.A Semyanistyi–Lizorkin space is introduced,on which the Radon transform is a bijection.We deal with the Radon transform on Q both by the Euclidean Fourier transform and the group Fourier transform.These two treatments are essentially equivalent.We also give an inversion formula by using wavelets,which does not require the smoothness of functions if the wavelet is smooth.In addition,we obtain an inversion formula of the Radon transform associated with the sub-Laplacian on Q.     

Wavelets and Hilbert Modules

A Hilbert C^*-module is a generalization of a Hilbert space for which the inner product takes its values in a C^*-algebra instead of the complex numbers. We use the bracket product to construct some Hilbert C^*-modules over a group C^*-algebra which is generated by the group of translations associated with a wavelet. We shall investigate bracket products and their Fourier transform in the space of square integrable functions in Euclidean space. We will also show that some wavelets are associated with Hilbert C^*-modules over the space of essentially bounded functions over higher dimensional tori.     

Resolvent Estimates Related with a Class of Dispersive Equations

We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers. The author was supported by the JSPS Grant-in-Aid for Scientific Research #17540140.     

Unitarity of Generalized Fourier–Gauss Transforms

Abstract

A generalized Fourier–Gauss transform is an operator acting in a Boson Fock space and is formulated as a continuous linear operator acting on the space of test white noise functions. It does not admit, in general, a unitary extension with respect to the norm of the Boson Fock space induced from the Gaussian measure with variance 1 but is extended to a unitary isomorphism if the Gaussian measure is replaced with the ones with different covariance operators. As an application, unitarity of a generalized dilation is discussed.     

Bessel identities in the waldspurger correspondence over the real numbers

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL ₂(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL ₂(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. Research of first author was partially supported by NSF grant DMS-0070762. Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.     

An over-determined boundary problem for the Helmholtz equation in a semiinfinite domain with a curvilinear boundary

In this paper we consider an over-determined Cauchy problem for the Helmholtz equation in a semiinfinite domain with a piecewise smooth curvilinear boundary. Applying the Fourier transform method in the space of distributions of slow growth, we establish the necessary and sufficient solvability conditions which connect the boundary functions. We construct integral representations of a solution.     

Explicit Hilbert spaces for certain unipotent representations III

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L²-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L²-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

Harmonic Analysis and Degenerate Diffusions on Euclidean Groups

We take up an idea, introduced by Chirikjian andKyatkin, of analyzing a family of left-invariant diffusion equations on Euclidean groups via the group Fourier transform. These diffusion equations model the probability distribution of the orientation in space of certain polymers in solution, including DNA. We study the evolution equations satisfied by the Fourier coefficients of such a solution. Our main task here is to estimate these Fourier coefficients sufficiently well that one canestimate the error in truncating the group inverse Fourier transform to afinite region.     

An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group

A classical result due to Paley and Wiener characterizes the existence of a non-zero function in \(L^2(\mathbb {R}),\) supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group M(n). We also relate this result to a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equation on M(n).     

Geometric aspects of frame representations of abelian groups

We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.