Гармонический анализ периодических на бесконечности функций в однородных пространствах

This paper is devoted to the study of slowly varying and almost periodic at infinity distributions from harmonic spaces; a number of spaces of homogeneous functions are considered. The notion of a harmonic space of distributions is introduced; this space is constructed by a homogeneous functional spaces. Properties of harmonic spaces of distributions endowed with the structure of Banach modules are studied. We prove that such spaces are isometrically isomorphic to the corresponding homogeneous functional spaces. Based on the definitions of slowly varying and almost periodic at infinity functions from a homogeneous space, we introduce the notions of slowly varying and almost periodic at infinity distributions from a harmonic space. Using methods of abstract harmonic analysis, we construct Fourier series of almost periodic distributions at infinity and examine their properties. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules.

In recent years, the use of Peter-Weyl theory (the theory of Fourier analysis on compact Lie groups) to define so-called “global symbols” of operators on compact Lie groups has emerged as a fruitful technique to study pseudo-differential operators. The aim of this thesis is to discuss similar techniques in the setting of compact homogeneous spaces. The approach is to relate operators on homogeneous spaces to those on compact Lie groups, and then to utilize the recently developed techniques on such groups. Two methods of associating operators on homogeneous spaces with those on compact Lie groups, called projective and horizontal lifting, along with their properties, merits and problems are considered. A key tool used in this analysis is the notion of a difference operator. This thesis includes a detailed study of such operators and their properties, combined with comprehensive calculations involving such operators on the homogeneous spaces Sn−1 = SO(n)/ SO(n− 1). This thesis concludes with a generalization of the symbolic calculus on compact Lie groups developed by M. Ruzhansky and V. Turunen together with a collection of conjectures, which if proven would relate the generalization to pseudo-differential theory on homogeneous spaces.

In this article we shall give a modern interpretation of transformation geometry. This subject has recently become of great interest to mathematics educators for use in kindergarten to high school, but has been paid too little attention at the college level. The usual approach to transformation geometry [5] or [12] consists of giving the classical geometries and then presenting their transformation groups. This certainly runs contrary to the ideas of the founder of transformation geometry, Felix Klein (1849-1929), who believed very strongly in a unified approach to mathematics whenever I shall adopt the view that the proper approach to transformation geometry is through the geometry of homogeneous spaces and that this affords a modern interpretation of Klein's program (the Erlanger Programm) as outlined in his original paper of 1872 [7]. Roughly speaking, Klein's program says that a geometry on a determines a group of transformations of the and that a group of transformations on the determines a geometry. While most modern mathematicians have been using the first idea (that very valuable information can be gained by looking at the group of transformations which leave the geometry invariant) we have neglected the second one. I claim that the theory of homogeneous spaces affords us a modern interpretation of the second point of Klein's program. I am not claiming that Klein foresaw the theory of homogeneous spaces but rather that with the help of the works of Riemann and E. Cartan (1869-1951) we can make precise (via the theory of homogeneous spaces) the notion that an arbitrarily chosen group will determine a geometry. If we are to adopt Klein's approach to transformations, what definition are we to take for geometry? There is certainly no easy answer to this question but the approach of G. F. B. Riemann (1826-1866) is certainly the most modern in spirit and is the one I shall use here. Riemann's inaugural address [111 begins: As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. The relationship between these presuppositions [the concept of space, and the basic properties of space] is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible. This means that we must first decide what space should be. In section 2 we define to be a homogeneous space; that is, the quotient of a topological group G by a subgroup L so that M = GIL. In section 3 we assume that G is a subgroup of the group of nonsingular matrices (that is, G and L are Lie groups). In this section we add some geometric structure to that of space, which we call a geometry on the homogeneous space, and so obtain the notion of lines. Throughout the last two sections of the paper we stress the fact that this definition of geometry includes in it, as special cases, Euclidean, Spherical and Hyperbolic geometries. We treat these three special cases in detail. (I do not claim, however, that this is the most general definition of geometry.) The first section of this paper gives a brief historical background before studying homogeneous spaces and their geometry. Where do homogeneous spaces belong in the realm of mathematics? They are not just used as an interpretation of Klein's Erlanger program; they are also used to do function theory (harmonic analysis [6]), to serve as models in differential geometry [8], and are used in mathematical physics [2]. It is as Klein remarks in the notes he added (in 1893) to the original Erlangen address ([7], p. 244): A model, whether constructed and observed or only vividly imagined, is for this geometry not a means to an end, but the subject itself.

Lie group representation theory and harmonic analysis on Lie groups and on their homogeneous spaces have built up enormous stores of knowledge as a significant and indispensable area of mathematics since the end of the World War II when the research in these areas began. These areas are closely interrelated with various other mathematical fields such as number theory, algebraic geometry, differential geometry, operator algebra, partial differential equations and mathematical physics, and are still growing and developing. In these circumstances, Ali Baklouti and Takaaki Nomura got a plan to organize joint seminars. The first seminar took place on November 2009 in Kerkennah Islands. This seminar was focused upon analysis with the keyword harmonic analysis as a core in such vast spread of research areas within the framework of academic program under the cooperation of Ministry of Higher Education, Scientific Research and Technology in Tunisia (MHESRT) and Japan Society for the Promotion of Science (JSPS). Many researchers on the forefront in both countries, experts in the concerned research areas from different corners, brought together the newest research results and discussed further progress of their research. This was a boosting reason for Ali Baklouti and Takaaki Nomura to continue the collaboration through the organization of a second Tunisian–Japanese meeting which took place in Sousse (December 2011) and is planned to emphasis direction axes of research toward commutative harmonic analysis (Fourier analysis on Euclidean spaces), analysis on homogeneous spaces, uncertainty principles and geometric analysis and some of their applications such as stochastic analysis and spectral theory. This seminar is also combined with an exchange based on the reciprocity program between Faculty of Mathematics, Kyushu University and Faculty of Science, Sfax University. In this sense, we list up post-doctoral students at Kyushu University, young researchers who graduated from Kyushu University, and young researchers in Sfax University. We also include young mathematicians from other than the two universities in the research area of non-commutative harmonic analysis. Thus we keep our eyes also on possible future exchanges by these young mathematicians, and the seminar was not a single one-time exchange between senior researchers. It is not so often that mathematicians who just made their debut

Polyakov spin factors and Laplacians on homogeneous spaces

The study of geodesic flows on homogenous spaces is an area of research that has yielded some fascinating developments. This book, first published in 2000, focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture. Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders. The treatment is as self-contained and elementary as possible. It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory

This article is a contribution to a Festschrift for S. Helgason. After a biographical sketch, we survey some of his research on several topics in geometric and harmonic analysis during his long and influential career. While not an exhaustive presentation of all facets of his research, for those topics covered we include reference to the current status of these areas. Preface Sigurður Helgason is known worldwide for his first book Differential Geometry and Symmetric Spaces. With this book he provided an entrance to the opus of Elie Cartan and Harish-Chandra to generations of mathematicians. On this the occa- sion of his 85th birthday we choose to reflect on the impact of Sigurður Helgason's sixty years of mathematical research. He was among the first to investigate system- atically the analysis of differential operators on reductive homogeneous spaces. His research on Radon-like transforms for homogeneous spaces presaged the resurgence of activity on this topic and continues to this day. Likewise he gave a geomet- rically motivated approach to harmonic analysis of symmetric spaces. Of course there is much more - eigenfunctions of invariant differential operators, propagation properties of differential operators, differential geometry of homogeneous spaces, historical profiles of mathematicians. Here we shall present a survey of some of these contributions, but first a brief look at the man.

A construction of relativistic wave equations on the homogeneous spaces of the Poincaré group is given for arbitrary spin chains. Parametrizations of the field functions and harmonic analysis on the homogeneous spaces are studied. It is shown that a direct product of Minkowski space time and two-dimensional complex sphere is the most suitable homogeneous space for the physical applications. The Lagrangian formalism and field equations on the Poincaré and Lorentz groups are considered. A boundary value problem for the relativistically invariant system is defined. General solutions of this problem are expressed via an expansion in hyperspherical functions defined on the complex two-sphere.

This paper presents estimation and detection techniques in homogeneous spaces that are optimal under the squared error loss function. The data is collected on a manifold which forms a homogeneous space under the transitive action of a compact Lie group. Signal estimation problems are addressed by formulating Wiener-Hopf equations for homogeneous spaces. The coefficient functions of these equations are the signal correlations which are assumed to be known. The resulting coupled integral equations on the manifold are converted to Wiener-Hopf convolutional integral equations on the group. These are solved using the Peter-Weyl theory of Fourier transform on compact Lie groups. The computational complexity of this algorithm is reduced using the bi-invariance of the correlations with respect to a stabilizer subgroup. The theory of matched filtering for isotropic signal fields is developed for signal classification where given a set of template signals on the manifold and a noisy test signal, the objective is to optimally detect the template buried in the test signal. This is accomplished by designing a filter on the manifold that maximizes the signal-to-noise-ratio (SNR) of the filtered output. An expression for the SNR is obtained as a ratio of quadratic forms expressed as Haar integrals over the transformation group. These integrals are expressed in the Fourier domain as infinite sums over the irreducible representations. Simplification of these sums is achieved by invariance properties of the signal function and the noise correlation function. The Wiener filter and matched filter are developed for an abstract homogeneous space and then specialized to the case of spherical signals under the action of the rotation group. Applications of these algorithms to denoising of 3D surface data, visual navigation with omnidirectional camera and detection of compact embedded objects in the stochastic background are discussed with experimental results.

We propose a method for calculating vacuum means of the scalar field energy-momentum tensor on Lie groups and homogeneous spaces. We use the generalized harmonic analysis based on the method of coadjoint representation orbits.

We develop a method for calculating vacuum expectation values of the energy-momentum tensor of a scalar field on homogeneous spaces with an invariant metric. Solving this problem involves the method of generalized harmonic analysis based on the method of coadjoint orbits.

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator $\mathscr L = -d/dt+A$ in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator $A$ and the semigroup generated by $A$. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart-Pr\"uss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.

This paper presents a systematic study for trigonometric polynomials over homogeneous spaces of compact groups. Let $H$ be a closed subgroup of a compact group $G$. Using the abstract notion of dual space $\\widehat{G/H}$, we introduce the space of trigonometric polynomials $\\mathrm{Trig}(G/H)$ over the compact homogeneous space $G/H$. As an application for harmonic analysis of trigonometric polynomials, we prove that the abstract dual space of anyhomogeneous space of compact groups separates points of the homogeneous space in some sense.

Let G be a semisimple Lie group and let if be a closed reductive subgroup of G.The homogeneous space X = G/H is called a semisimple homogeneous space.A fundamental goal of harmonic analysis is to understand the group action of G on the various function spaces of X.In particular, the L 2 -harmonic analysis on X is to decompose L 2 (X) as a direct integral of irreducibles, i.e., to find a family of irreducible unitary representations {V ω \ ω € Ω} of G, and a measure v on the set Ω, so that JΩThe above decomposition is called the Plancherel formula for the homogeneous space X.In this paper we prove the Plancherel formulae for some non-symmetric semisimple homogenous spaces.

Vector-valued functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity are investigated. The concept of the Fourier series of a function (distribution), periodic or almost periodic at infinity, with coefficients that are functions (distributions) slowly varying at infinity, is introduced. The properties of the Fourier series are investigated and an analogue of Wiener’s theorem on absolutely convergent Fourier series is obtained for functions periodic at infinity. Special attention is given to criteria ensuring that solutions of differential or difference equations are periodic or almost periodic at infinity. The central results involve theorems on the asymptotic behaviour of a bounded operator semigroup whose generator has no limit points on the imaginary axis. In addition, the concept of an asymptotically finite-dimensional operator semigroup is introduced and a theorem on the structure of such a semigroup is proved. Bibliography: 39 titles.