Platonic polyhedra tune the 3-sphere: harmonic analysis on simplices

The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called “classical separation of variables theorem” for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

Introduction Representations of $\mathbf {SU}(2)$ The general theory of linear representations of compact groups Lie theory of representations of compact connected Lie groups Induced representations of compact groups Spherical functions on compact groups Examples spherical harmonics The general theory of spherical functions Fourier and Plancherel transforms Extension of the Plancherel transform The subtleties of harmonic analysis Differential properties of spherical functions on Lie groups Spherical functions on semisimple Lie groups More on $\mathbf {SL}(2, \mathbf R)$ Automorphic functions Groups of isometries and Bessel functions Other special functions References.

A method for measuring the form error of spherical parts was developed in the previous report. The measured results showed that steel balls used as ball bearing assemblies usually had the form error of three or four undulations. In the present report, a harmonic analysis by the spherical function is introduced to evaluate the sphericities of the steel balls. Dimensional spectra have invariant values for each ball even if the measuring coordinate axes are changed. Steel balls which are sampled in various states of different finishing processes are measured, and the dimensional spectra are calculated. The calculated results show that the dimensional spectra of steel balls that are in the same state of the finishing process have the same values and that the 3rd-order spectrum has a tendency to be left behind selectively by the polishing process. Evidence has been given that the harmonic analysis by the spherical function is successful for evaluating the sphericity of steel balls.

THE need for a treatise on harmonic analysis, more particularly on spherical harmonics, has been recognised for a long time by mathematicians who teach the subject or who find it necessary to use spherical harmonics in their investigations. Those who had the good fortune to be introduced to the study of the subject by the author of the present work had always hoped that he would find time to undertake the preparation and publication of such a treatise. It may be said at once that the work now published fulfils the expectations they had formed. Although many books which contain information about spherical harmonics have appeared since the publication of Heine's “Kugelfune-tionen”, none of them has claims to be considered a complete treatise on the subject, and Heine's work, though it has proved of great service to students of the subject and to mathematicians who had occasion to use harmonic analysis in their investigations, is a compilation of results which contains the material for a treatise rather than a treatise. Much has been written on the subject since the appearance of Heine's two volumes. The mere perusal of the literature involves much labour, and the task of selecting from this mass of literature what is of prime importance to students of the subject and what is necessary for the mathematical physicist is one which demands a wide knowledge of the related branches of mathematics and of applied mathematics. That the author has fulfilled these requirements the present treatise is the evidence. The first four chapters give a complete account of the origin of the ordinary spherical harmonics; Legendre's coefficients of the first and second kinds and the associated functions are treated in detail, their more important properties are developed, and their expression in the form of series and of definite integrals is investigated. Solid harmonics are discussed fully, the addition theorem for ordinary spherical harmonics is established, and the connexion of the functions with potential theory is dealt with. The results given in this part of the treatise are sufficient for the solution of the problems of potential theory and of other branches of mathematical physics where the space under consideration is the space inside or outside the surface of a sphere or the space between two concentric spheres.

Source localization has been studied in the spatial domain using differential geometry in earlier work. However, parameters of the sensor array manifold have hitherto not been investigated for source localization in spherical harmonics domain. The objective of this work is to represent and model the manifold surface using differential geometry. The system model for source localization over a spherical harmonic manifold is first formulated. Subsequently, the manifold parameters are modeled in the spherical harmonics domain. Source localization methods using MUSIC and MVDR over the spherical harmonics manifold are developed. Experiments on source localization using a spherical microphone array indicate high resolution in noise.

Harmonic analysis on [formula omitted]: Part II

The paper presents new developments in harmonic analysis associated with the motion transformations embedded in digital signals. In this context, harmonic analysis provides motion analysis with a complete theoretical construction of perfectly matching concepts and a related toolbox leading to fast algorithms. This theory can be built from only two assumptions: an associative structure for the local motion transformations expressed as a Lie group and a principle of optimality for the global evolution expressed as a variational extremal. Motion analysis means not only detection, estimation, interpolation, and tracking but also propagator's motion-compensated filtering, signal decomposition, and selective reconstruction. The optimality principle defines the trajectory and provides the appropriate equations of motion, the selective tracking equations, the selective constants of motion to be tracked, and all the symmetries to be imposed on the system. The harmonic analysis provides new special functions, orthogonal bases, partial differential equations (PDE), ordinary differential equations (ODE) and integral transforms. The tools to be developed rely on group representations, continuous and discrete wavelets, the estimation theory (prediction, smoothing and interpolation) and filtering theory (Kalman filters, motion-based convolutions, integral transforms). All the algorithms are supported by fast and parallelizable implementations based on the fast Fourier transform (FFT) and dynamic programming.

Over the past several years, operator algebraists have become increasingly interested in the problem of calculating the K-theory of group C∗-algebras. The focal point of research in this area ist he BaumConnesConjecture[BCH],which proposes a description of K-theory for the C∗-algebra of a group in terms of homology and the representation theory of compact subgroups. Although the main applications of the Baum-Connes Conjecture are to issues in geometry and topology, the conjecture also appears to be of interest from the point of view of harmonic analysis. Whereas for applications to topology one is concerned with discrete groups G (arising as the fundamental groups of manifolds), the conjecture’s links with harmonic analysis appear to be the strongest for reductive Lie groups and p-adic groups. The purpose of these notes is to convey to a reasonably broad audience some byproducts of the authors’ research into the C∗-algebra K-theory of the p-adic group GL(N), which culminated in a proof of the Baum-Connes Conjecture in this case [BHP2]. Along the way to the proof a number of interesting issues came to light which we feel deserve some exposure, even though our understanding of them is far from complete, and is indeed mostly very tentative. Much of what follows is focused on what we call here chamber homology, which is a type of equivariant homology associated to the action of a reductive p-adic group on its Bruhat-Tits affine building. The problem of computing chamber homology can be approached from a number of different directions. An especially interesting problem is to reconcile chamber homology with the Bernstein decomposition for representations of reductive p-adic groups [Be, BD]. This appears to be a far from trivial matter, even in comparatively simple cases. In Section 5 we formulate two very general conjectures which give a broad description of a Bernstein decomposition in chamber homology (perhaps we should call our conjectures questions, since the evidence we have gathered in their favor is not overwhelming).

We construct non-trivial elements of order 2 in the homotopy groups $\pi_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $\Omega^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $\pi_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-invariant. As consequences, we (a) obtain non-trivial elements of order 2 in $\pi_* Diff(D^m,\partial)$ for m greater or equal to 6, and * + m congruent 0 or 1 modulo 8; (b) these elements remain non-trivial in $\pi_* Diff(M)$ where M is a closed spin manifold of the same dimension m and * > 0; (c) they act non-trivially on the corresponding homotopy group of the space of metrics of positive scalar curvature of such an M; in particular these homotopy groups are all non-trivial. The same applies to all other diffeomorphism invariant metrics of positive curvature, like the space of metrics of positive sectional curvature, or the space of metrics of positive Ricci curvature, provided they are non-empty. Further consequences are: (d) any closed spin manifold of dimension m greater or equal to 6 admits a metric with harmonic spinors; (e) there is no analogue of the odd-primary splitting of $(PL/O)_{(p)}$ for the prime 2; (f) for any $bP_{8j+4}$-sphere (where j > 0) of order which divides 4, the corresponding element in $\pi_0 Diff(D^{8j+2},\partial)$ lifts to $\pi_{8j-4} Diff(D^6,\partial)$, i.e., lies correspondingly deep down in the Gromoll filtration.

Spherical cap harmonic analysis has become a well-known technique for regional modelling of fields that can be expressed as the gradient of a scalar potential, such as for example the geomagnetic field and its secular variation. Up to now, the method has been regularized by a purely statistical technique: coefficients that are considered to be statistically insignificantly small are simply set to zero. This method lacks physical justification and ignores resolution; individual coefficients may be small but well-resolved, while coefficients with a large value may be poorly resolved. We implement the more physical regularization technique of minimizing a certain feature of the field, for example the field strength over the cap surface, analogous to the regularization techniques widely used in global spherical harmonic analysis. The mathematical difference between spherical cap harmonics (SCHA) and global spherical harmonics analyses (SHA) lies in the basis functions. While these are completely orthogonal in SHA, this is not the case in SCHA. This leads to the existence of certain linear combinations of coefficients that hardly contribute to the field, which makes the statistical rejection criterion meaningless. With the physical regularization the individual coefficients become meaningful, as we show by modelling the secular variation from a data set of 30 years of European observatory measurements, repeat station and ground vector surveys.

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation theoretic construction $ϕ(x) = $ where $e$ is a $K_\infty$--fixed unit vector for $π$. Specifically, we look at representations $π_\infty = \varinjlim π_n$ of $G_\infty$ where $π_n$ is $K_n$--spherical, so the spherical representations $π_n$ and the corresponding spherical functions $ϕ_n$ are related by $ϕ_n(x) = $ where $e_n$ is a $K_n$--fixed unit vector for $π_n$, and we consider the possibility of constructing a $K_\infty$--spherical function $ϕ_\infty = \lim ϕ_n$. We settle that matter by proving the equivalence of condtions (i) $\{e_n\}$ converges to a nonzero $K_\infty$--fixed vector $e$, and (ii) $G_\infty/K_\infty$ has finite symmetric space rank (equivalently, it is the Grassmann manifold of $p$--planes in $\F^\infty$ where $p < \infty$ and $\F$ is $\R$, $\C$ or $\H$). In that finite rank case we also prove the functional equation $ϕ(x)ϕ(y) = \lim_{n\to \infty} \int_{K_n}ϕ(xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.

Various new nonembeddability results (mainly into L/sub 1/) are proved via Fourier analysis. In particular, it is shown that the edit distance on {0, 1}/sup d/ has L/sub 1/ distortion (log d)/sup 1/2 - o(1)/. We also give new lower bounds on the L/sub 1/ distortion of quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.

view Abstract Citations (54) References (34) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Quasi-rigid Rotation of Coronal Magnetic Fields Wang, Y. -M. ; Sheeley, N. R., Jr. ; Nash, A. G. ; Shampine, L. R. Abstract Assuming that the coronal magnetic field can be approximated by a current-free extension of the photospheric field, the authors use spherical harmonic analysis and numerical simulations to study its rotational properties. In the outer corona, they find that the rotation rate is determined by three principal factors: 1. "Coronal filtering". 2. Global averages of the photospheric rotation rate. 3. Ongoing source eruptions. These principles allow to understand the observationally inferred rotational properties of the outer coronal field. The overall rigidity of the rotation profile reflects the tendency for the photosphere's non-axisymmetric flux to be concentrated toward lower latitudes, where the rotational shear is small; increased curvature and asymmetry occur during the rising phase of the sunspot cycle because of the presence of higher latitude flux. The coronal rotation rate shows a progressive acceleration due to the equatorward migration of sunspots. Publication: The Astrophysical Journal Pub Date: April 1988 DOI: 10.1086/166205 Bibcode: 1988ApJ...327..427W Keywords: Rigid Rotors; Solar Corona; Solar Magnetic Field; Solar Rotation; Harmonic Analysis; Photosphere; Solar Activity; Spherical Harmonics; Solar Physics; SUN: CORONA; SUN: MAGNETIC FIELDS; SUN: ROTATION full text sources ADS |

The question is pursued of whether there exists acomplete alternate formulation of classical electrodynamics in terms of asingle scalar function, which may then lead to a new formulation of quantum electrodynamics. Using harmonic analysis and spin-weighted spherical harmonics we study such a theory and solve it for three deliberately simple but characteristic problems: point charges, current loops and antennae. The connection between the field strength and potential formulations is established, as is the relation of this theory to the conventional one using vector spherical harmonics.

Section 0 Lot G be a connected noncompact semisimple Lie group with finite center and real rank one; fix a maximal compact subgroup K. Our concern in this paper is Fourier analysis on the Riemannian symmetric space G]K. We shall analyze the local and global behavior of spherical functions, the boundedness of multiplier operators, and the inversion of differential operators. The core of the paper, however, is an analysis of how the size of a function is controlled by the size of its Fourier transform. There is an extensive literature on such topics, centered about the Paley-Wiener and Plancherel theorems. Our work relies heavily on these earlier ideas and techniques, to which detailed reference will be made in the body of the paper. The problems we wish to solve, however, require estimates more precise and of a different nature than are necessary for the Plancherel or Paley-Wienor theorem. Thus the first two sections of this paper are devoted to the construction of various asymptotic expansions for spherical functions; in later sections we show how these expansions may be applied to the Fourier analysis of G/K.