Nonstationary Wavelets on the m-Sphere for Scattered Data

We consider approximation methods defined by translates of a positive definite function on a compact group. A characterization of the native space generated by a positive definite function on a compact group is presented. Starting from Bochner's theorem, we construct examples of well-localized positive definite central functions on the rotation group (3). Finally, the stability of the interpolation problem and the error analysis for the given examples are studied in detail.

Let S be a subset of an amenable group G such that e ∈ S and S–1 = S. The main result of this paper states that if the Cayley graph of G with respect to S has a certain combinatorial property, then every positive definite operator-valued function on S can be extended to a positive definite function on G. Several known extension results are obtained as corollaries. New applications are also presented.

view Abstract Citations (12) References (23) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Second-Order Perturbation Theory in an Expanding Universe: Spherical Harmonics Expansion Zaroubi, Saleem ; Hoffman, Yehuda Abstract The formalism of gravitational instability in an expanding universe is developed here to second order in terms of the spherical harmonics representation and is applied to study the evolution of single localized perturbations to an otherwise flat Einstein-de Sitter universe. A simple analytical form of the radial dependence of the various harmonics is used to model the structure expected for a scale-free Gaussian perturbation field. The general formalism is applied to study the dynamics of the monopole terms, i.e., the spherically averaged variables, of a proto- object made of monopole and quadrupole terms only. Our main result is that at the inner part, i.e., the core, the dynamics is dominated by the monopole term, and it reproduces the exact spherical top-hat model. However, the outer region, i.e., the halo, is dominated by the quadrupole terms, and its dynamics differs significantly from the top-hat model. The l = 2 terms lead to the formation of a secondary maximum in the run of the density (averaged over spherical shells) versus radius. This is a pure second-order effect whose very clear signature is that it occurs at a radius of the transition to the nonlinear regime. Indeed, such a secondary maximum has been observed in the outer parts of real clusters of galaxies and in N-body numerical simulations. Another pure second- order effect discovered here is the case of a pure quadrupole primordial perturbation. For such a perturbation the vanishing monopole does not grow in the linear regime, but is unstable to second order. We find a considerable growth for typical values of parameters, and conjecture that this is the main dynamical factor in the formation of filaments in the large-scale structure of the universe. Publication: The Astrophysical Journal Pub Date: September 1993 DOI: 10.1086/173052 Bibcode: 1993ApJ...414...20Z Keywords: Cosmology; Galactic Clusters; Hubble Constant; Perturbation Theory; Spherical Harmonics; Computational Astrophysics; Digital Simulation; Universe; Astrophysics; COSMOLOGY: THEORY; GALAXIES: CLUSTERING; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE full text sources ADS |

Fourier–Stieltjes Algebras of Locally Compact Groupoids

Stability results for approximation by positive definite functions on SO(3)

Strictly positive definite functions by integral transforms

Diagonalization of the fast multipole method (FMM) for the Helmholtz equation is usually achieved by expanding the multipole representation in propagating plane waves. The resulting k-space integral over the Ewald sphere is numerically evaluated. Storing the k-space quadrature samples of the method of moments (MoM) basis functions constitutes a large portion of the overall memory requirements of the resulting algorithm for solving the integral equations of scattering and radiation problems. In this paper, it is proposed to expand the k-space representation of the basis functions by spherical harmonics in order to reduce the sampling redundancy introduced by numerical quadrature rules. Aggregations, plane wave translations, and disaggregations in the realized multilevel fast multipole method (MLFMM) are carried out using the k-space samples of a numerical quadrature rule. However, the incoming plane waves on the finest MLFMM level are expanded in spherical harmonics again. Thus, due to the orthonormality of spherical harmonics, the testing integrals for the individual testing functions are simplified into series over products of spherical harmonics expansion coefficients. Overall, the resulting MLFMM can save a considerable amount of memory without compromising accuracy and numerical speed.

The main purpose of this paper is an explicit construction of generalized Gaussian process with function tb(V) = bH(V), where H(V) = n - h(V), h(V) is the number of singletons in a pair-partition V ∈ 𝒫2(2n). This gives another proof of Theorem of A. Buchholtz15 that tb is positive definite function on the set of all pair-partitions. Here there are some new combinatorial formulas presented. Connections with free additive convolutions probability measure on ℝ are also done. There are new positive definite functions on permutations presented. What is more, it is proven that the function H is norm (on the group S(∞) = ⋃S(n)).

For any vector r = r1 + r2 an expansion is derived for the product of a power rN of its magnitude and a surface spherical harmonic YLM(ϑ, φ) of its polar angles in terms of spherical harmonics of the angles (ϑ1, φ1) and (ϑ2, φ2). The radial factors satisfy simple differential equations; their solutions can be expressed in terms of hypergeometric functions of the variable (r</r>)2, and the leading coefficients by means of Gaunt's coefficients or 3j symbols. A number of linear transformations and three-term recurrence relations between the radial function are derived; but in contrast to the case L = 0, no generally valid expressions symmetric in r1 and r2 could be found. By interpreting the terms operationally, an expansion is derived for the product of YLM(ϑ, φ) and an arbitrary function f(r). The radial factors are expansions in derivatives of f(r>); for spherical waves, they factorize into Bessel functions of r1 and r2 in agreement with the expansion by Friedman and Russek. The 3j symbols are briefly discussed in an unnormalized form; the new coefficients are integers, satisfying a simple recurrence relation through which they can be arranged on a five-dimensional generalization of Pascal's triangle.

In this paper we study strictly positive definite functions on the unit sphere of the m m -dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977–981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de Boor-Ron general “least solution for the multivariate polynomial interpolation problem”. The second, which is suitable only for m = 2 m=2 , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.

A number of consistent models describing stationary positive definite functions on stem from Bochner’s celebrated theorem characterizing continuous and stationary positive definite functions on If denotes the unit sphere in the same is true of positive definite functions on which are radial with respect to the component and stationary with respect to the component. In this paper, we summarize results on these topics, mainly those that somehow characterize the positive definiteness of the function through Fourier transforms of the sections of the function itself. We present a new perspective on the existent results in the literature along with new characterizations and applications.

Introduction. 1.1 Linear Time-Invariant Systems. 1.2 Nonlinear Systems. 1.3 Equilibrium Points. 1.4 First-Order Autonomous Nonlinear Systems. 1.5 Second-Order Systems: Phase-Plane Analysis. 1.6 Phase-Plane Analysis of Linear Time-Invariant Systems. 1.7 Phase-Plane Analysis of Nonlinear Systems. 1.8 Higher-Order Systems. 1.9 Examples of Nonlinear Systems. 1.10 Exercises. Mathematical Preliminaries. 2.1 Sets. 2.2 Metric Spaces. 2.3 Vector Spaces. 2.4 Matrices. 2.5 Basic Topology. 2.6 Sequences. 2.7 Functions. 2.8 Differentiability. 2.9 Lipschitz Continuity. 2.10 Contraction Mapping. 2.11 Solution of Differential Equations. 2.12 Exercises. Lyapunov Stability I: Autonomous Systems. 3.1 Definitions. 3.2 Positive Definite Functions. 3.3 Stability Theorems. 3.4 Examples. 3.5 Asymptotic Stability in the Large. 3.6 Positive Definite Functions Revisited. 3.7 Construction of Lyapunov Functions. 3.8 The Invariance Principle. 3.9 Region of Attraction. 3.10 Analysis of Linear Time-Invariant Systems. 3.11 Instability. 3.12 Exercises. Lyapunov Stability II: Nonautonomous Systems. 4.1 Definitions. 4.2 Positive Definite Functions. 4.3 Stability Theorems. 4.4 Proof of the Stability Theorems. 4.5 Analysis of Linear Time-Varying Systems. 4.6 Perturbation Analysis. 4.7 Converse Theorems. 4.8 Discrete-Time Systems. 4.9 Discretization. 4.10 Stability of Discrete-Time Systems. 4.11 Exercises. Feedback Systems. 5.1 Basic Feedback Stabilization. 5.2 Integrator Backstepping. 5.3 Backstepping: More General Cases. 5.4 Examples. 5.5 Exercises. Input-Output Stability. 6.1 Function Spaces. 6.2 Input-Output Stability. 6.3 Linear Time-Invariant Systems. 6.4 Lp Gains for LTI Systems. 6.5 Closed Loop Input-Output Stability. 6.6 The Small Gain Theorem. 6.7 Loop Transformations. 6.8 The Circle Criterion. 6.9 Exercises. Input-to-State Stability. 7.1 Motivation. 7.2 Definitions. 7.3 Input-to-State Stability (ISS) Theorems. 7.4 Input-to-State Stability Revisited. 7.5 Cascade Connected Systems. 7.6 Exercises. Passivity. 8.1 Power and Energy: Passive Systems. 8.2 Definitions. 8.3 Interconnections of Passivity Systems. 8.4 Stability of Feedback Interconnections. 8.5 Passivity of Linear Time-Invariant Systems. 8.6 Strictly Positive Real Rational Functions. Exercises. Dissipativity. 9.1 Dissipative Systems. 9.2 Differentiable Storage Functions. 9.3 QSR Dissipativity. 9.4 Examples. 9.5 Available Storage. 9.6 Algebraic Condition for Dissipativity. 9.7 Stability of Dissipative Systems. 9.8 Feedback Interconnections. 9.9 Nonlinear L2 Gain. 9.10 Some Remarks about Control Design. 9.11 Nonlinear L2-Gain Control. 9.12 Exercises. Feedback Linearization. 10.1 Mathematical Tools. 10.2 Input-State Linearization. 10.3 Examples. 10.4 Conditions for Input-State Linearization. 10.5 Input-Output Linearization. 10.6 The Zero Dynamics. 10.7 Conditions for Input-Output Linearization. 10.8 Exercises. Nonlinear Observers. 11.1 Observers for Linear Time-Invariant Systems. 11.2 Nonlinear Observability. 11.3 Observers with Linear Error Dynamics. 11.4 Lipschitz Systems. 11.5 Nonlinear Separation Principle. Proofs. Bibliography. List of Figures. Index.

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

Extensions of positive definite functions on free groups

An intriguing and deep analogy between classical and quantum states is revealed using the notion of positive definite function. By Bochner's theorem, (classical) positive definite functions on phase space can be obtained by taking the Fourier transform of probability measures; similarly, quantum positive definite functions are the image via the Fourier-Plancherel operator of Wigner quasi-probability distributions. Considering the basic properties of positive definite functions — classical and quantum — one is led to define a class of semigroups of operators, the so-called classical-quantum semigroups. It is then natural to wonder whether they have any physical meaning. It turns out that the classical-quantum semigroups can also be obtained by dequantizing a certain class of quantum dynamical semigroups, namely, the classical noise semigroups. This correspondence fits in a more general group-theoretical framework in which a larger class of quantum dynamical semigroups, the twirling semigroups, can be suitably dequantized. Connections with quantum information science will be briefly discussed.