A KAM Result on Compact Lie Groups

Acta Applicandae Mathematicae - We describe some recent results on existence of quasi-periodic solutions of Hamiltonian PDEs on compact manifolds. We prove a linear stability result for the...     

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms

The polar decomposition, a well-known algorithm for decomposing real matrices as the product of a positive semidefinite matrix and an orthogonal matrix, is intimately related to involutive automorphisms of Lie groups and the subspace decomposition they induce. Such generalized polar decompositions, depending on the choice of the involutive automorphism σ , always exist near the identity although frequently they can be extended to larger portions of the underlying group. In this paper, first of all we provide an alternative proof to the local existence and uniqueness result of the generalized polar decomposition. What is new in our approach is that we derive differential equations obeyed by the two factors and solve them analytically, thereby providing explicit Lie-algebra recurrence relations for the coefficients of the series expansion. Second, we discuss additional properties of the two factors. In particular, when σ is a Cartan involution, we prove that the subgroup factor obeys similar optimality properties to the orthogonal polar factor in the classical matrix setting both locally and globally, under suitable assumptions on the Lie group G . September 12, 2000. Final version received: April 16, 2001.     

Brownian Processes for Monte Carlo Integration on Compact Lie Groups

This article proposes a Monte Carlo approach for the evaluation of integrals of smooth functions defined on compact Lie groups. The approach is based on the ergodic property of Brownian processes in compact Lie groups. The article provides an elementary proof of this property and obtains the following results. It gives the rate of almost sure convergence of time averages along with a “large deviations” type upper bound and a central limit theorem. It derives probability of error bounds for uniform approximation of the paths of Brownian processes using two numerical schemes. Finally, it describes generalization to compact Riemannian manifolds.     

A Generalized Beurling Theorem for Some Lie Groups

Mathematical Notes - For any real numbers p,q ≥ 1, we present in this paper a (p, q)-generalized version of Beurling’s uncertainty principle for ?n, which largely extends the...     

Asymptotic Expansions of Hermite Functions on Compact Lie Groups

Let G be a compact, connected Lie group endowed with a bi-invariant Riemannian metric. Let _t be the heat kernel on G; that is, _t is the fundamental solution to the heat equation on the group determined by the Laplace–Beltrami operator. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new family of functions on G. These functions, obtained from _t and its derivatives, are the compact group analogs of the classical Hermite polynomials on . Previous work of this author has established that these Hermite functions approach the classical Hermite polynomials on in the limit of small t, where the Hermite functions are viewed as functions on via composition with the exponential map. The present work extends these results by showing that these Hermite functions can be expanded in an asymptotic series in powers of . For symmetrized derivatives, it is shown that the terms with fractional powers of t vanish. Additionally, the asymptotic series for Hermite functions associated to powers of the Laplacian are computed explicitly. Remarkably, these asymptotic series terminate, yielding a polynomial in t.     

The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane

We consider the problem of reconstructing a function on the disk from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk under the additional condition that the curves that do not meet coincide with the corresponding straight lines.     

The Elementary Geometric Structure of Compact Lie Groups

We give geometric proofs of some of the basic structure theoremsfor compact Lie groups. The goal is to take a fresh look atthese theorems, prove some that are difficult to find in theliterature, and illustrate an approach to the theorems thatcan be imitated in the homotopy theoretic setting of p-compactgroups. 1991 Mathematics Subject Classification 22E15, 55P35.     

Nuclear Pseudo-Differential Operators in Besov Spaces on Compact Lie Groups

In this work we establish the metric approximation property for Besov spaces defined on arbitrary compact Lie groups. As a consequence of this fact, we investigate trace formulae for nuclear Fourier multipliers on Besov spaces. Finally, we study the r-nuclearity, the Grothendieck–Lidskii formula and the (nuclear) trace of pseudo-differential operators in generalized Hörmander classes acting on periodic Besov spaces. We will restrict our attention to pseudo-differential operators with symbols of limited regularity.     

Ergodic Actions of Semisimple Lie Groups on Compact Principal Bundles

Let G = SL(n, ?) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/Λ is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H′Λ′. Consequently, there is a strong restriction on the groups K that can arise over this particular M.     

Finite Subgroups of Compact Lie Groups and Maximal Exponents of Lie Algebras

In this paper, the definition of the maximal exponent for Lie algebras corresponding to compact simple Lie groups is given. For this purpose, the imbeddings of finite groups by regular elements in compact Lie groups are used. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

Generalized Group Algebras of Locally Compact Groups

This article studies the homological properties of generalized group algebra L ¹(G, A) of a locally compact group G over a Banach algebra A with an identity of norm 1. It is shown that if L ¹(G, A) is right continuous then G is finite and A is right continuous. It is also shown that L ¹(G, A) is right self-injective if and only if G is finite and A is right self-injective.     

Derivative of the Wiener Measure on Trajectories in Compact Lie Groups

Mathematical Notes -     

Nilpotient Lie群上Herz空间及它的某些应用

§1 .　IntroductionLetGbeaconnectedandsimplyconnectednilpotentLiegroupwith(bi invariant)HaarmeasuredgandLiealgebraG .Theexponetialmapissurjectiveby [12 ],Theorem 3.6 .1.Onecanassociatedasubellipticdistance (g ,h)d′( g ;h)witheachfixedalgeraicbasisa1 ,a2 ,…ad′ofG .Foralli∈ { 1,2 ,… ,d′} ,letAi =dL(ai)denotethegeneratorsoflefttranslationsactingontheclass{ai} .Thisdistancehasthecharacterizationd′( g ;h) =sup{ ｜ψ( g) - ψ(h) ｜ :ψ∈C∞0 (G) ,∑d′i=1｜ (Aiψ) ｜2 ≤ 1} ,(see [9],…