Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Better lower bounds on detecting affine and spherical degeneracies

We show that in the worst case, Ω(n ^d ) sidedness queries are required to determine whether a set ofn points in ℝ ^d is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing Ω(n ^d ) “collapsible” simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an Ω(n ^d ) lower bound on the number of sidedness queries required to determine the order type of a set ofn points in ℝ ^d . Using similar techniques, we also show that Ω(n ^d+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set ofn points in ℝ ^d . An earlier version of this paper was presented at the 34th Annual IEEE Symposium on Foundations of Computer Science [8]. This research has been supported by NSF Presidential Young Investigator Grant CCR-9058440.     

On the second derivative of the spherical contact distribution function of smooth grain models

We consider a stationary grain model Ξ in ℝ ^d with convex, compact and smoothly bounded grains. We study the spherical contact distribution function F of Ξ and derive (under suitable assumptions) an explicit formula for its second derivative F″. The value F″(0) is of a simple form and admits a clear geometric interpretation.For the Boolean model we obtain an interesting new formula for the(d− 1)-st quermass density. Received: 22 November 1999 / Revised version: 2 November 2000 /?Published online: 14 June 2001     

Convolution with oscillating kernels: Bochner-Riesz and spherical partial sums operators

The paper is an up to date survey of results on theL ^p boundedness, 1 ≤p ≤ ∞, of convolution operators with oscillating kernels, mainly Bochner-Riesz and the spherical partial sums operators in ℝ ⁿ ,n ≥ 2 and their maximal operators. Consequences are norm convergence and almost everywhere convergence of Fourier series. Il lavoro è una rassegna di risultati, fino ai più recenti, sulla limitatezzaL ^p , 1 ≤p ≤ ∞, di operatori di convoluzione con nuclei oscillanti, con particolare riguardo agli operatori di Bochner-Riesz e agli operatori somme parziali sferiche. Come applicazione si hanno teoremi di convergenza in norma e quasi ovunque per serie di Fourier.

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(Conferenza tenuta il 27 giugno 1988)     

Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

Extremum problems for entire functions of exponential spherical type

We consider extremum problems for entire functions of exponential spherical type related to important extremum problems on the optimal point (the Chernykh point) in the sharp jackson inequality in the spaceL ₂(ℝ ⁿ ) and the connection between codes and designs on the torusT ⁿ . Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 179–187, August, 2000.     

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions on higher rank complex simple groups, we establish estimates for which are of the form in the spectral parameter and have uniform exponential decay in regular directions in the group variable a _t . Here is an explicit constant depending on G, and may be singular, for instance.?The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L ^p for singular sphere averages on complex semisimple Lie groups for all p in . We use these to establish singular differentiation theorems and pointwise ergodic theorems in L ^p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions. Submitted: May 2000, Revised version: October 2000.     

About spherical functions onSL(2,ℝ)

In this paper, we will discuss some properties of the (n, m)-spherical functions on the Lie groupG = SL(2,ℝ), and obtain the decomposition off inC _c ⁴ (G) into these functions. Also we give the Fourier inversion formula for the (n, m)-spherical functions inC _c ³ (G).     

An optimal theorem for the spherical maximal operator on the Heisenberg group

Let be the Heisenberg group and μ _r be the normalized surface measure on the sphere of radiusr in ℂ ⁿ . Let . We prove an optimalL ^p-boundedness result for the spherical maximal functionMf, namely we prove thatM is bounded onL ^p(I ⁿ ) if and only ifp>2n/2n−1.     

On the number of affine equivalence classes of spherical tube hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} that are (k, n − k)-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n − k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n − 2, n ≥ 7; (ii) k = n − 3, n ≥ 7; (iii) k ≤ n − 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels–Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels–Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.     

A generalization of the fundamental theorem of spherical harmonic theory

We consider a boundary-value problem of the first kind for a self-adjoint differential operator with constant coefficients on a domain in ℝⁿ bounded by an ellipsoid; boundary conditions are defined by an arbitrary polynomial of degree N. It is proved that the solution of the problem is again a polynomial of degree ≤N. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

Central limit theorem for integrated square error of kernel estimators of spherical density

LetX ₁,…,X _n be iid observations of a random variableX with probability density functionf(x) on the q-dimensional unit sphere Ω_q in R^q+1,q ⩾ 1. Let be a kernel estimator off(x). In this paper we establish a central limit theorem for integrated square error off _n under some mild conditions.     

A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines

We present an overlapping domain decomposition technique for solving the hypersingular integral equation on the sphere with spherical splines. We prove that the condition number of the additive Schwarz operator is bounded by O(H/δ), where H is the size of the coarse mesh and δ is the overlap size, which is chosen to be proportional to the size of the fine mesh. In the case that the degree of the splines is even, a better bound O(1 + log²(H/δ)) is proved. The method is illustrated by numerical experiments on different point sets including those taken from magsat satellite data.     

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney, Tremaine in Galactic Dynamics, Princeton University Press, Princeton, 1987) is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov (Sov. Astron. 4:859–867, 1961). In the previous work (Lemou et al. in A new variational approach to the stability of gravitational systems, submitted, 2011), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature (Lynden-Bell in Mon. Not. R. Astron. Soc. 144:189–217, 1969; Gardner in Phys. Fluids 6:839–840, 1963; Wiechen et al. in Mon. Not. R. Astron. Soc. 223:623–646, 1988; Aly in Mon. Not. R. Astron. Soc. 241:15, 1989). In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.     

McLaren’s improved snub cube and other new spherical designs in three dimensions

Evidence is presented to suggest that, in three dimensions, spherical 6-designs withN points exist forN=24, 26,≥28; 7-designs forN=24, 30, 32, 34,≥36; 8-designs forN=36, 40, 42,≥44; 9-designs forN=48, 50, 52,≥54; 10-designs forN=60, 62, ≥64; 11-designs forN=70, 72,≥74; and 12-designs forN=84,≥86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and—although not identified as such by McLaren—consists of the vertices of an “improved” snub cube, obtained from Archimedes' regular snub cube (which is only a 3-design) by slightly shrinking each square face and expanding each triangular face. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5 designs exist forN=12, 16, 18, 20,≥22. It is conjectured, albeit with decreasing confidence fort≥9, that these lists oft-designs are complete and that no other exist. One of the constructions gives a sequence of putative sphericalt-designs withN=12m points (m≥2) whereN=1/2t ²(1+o(1)) ast→∞.     

The Coulomb energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n-designs, where a spherical n-design is a set of m points on the unit sphere S ² ⊂ ℝ³ that gives an equal weight cubature rule (or equal weight numerical integration rule) on S ² which is exact for spherical polynomials of degree ⩽ n. (A sequence Ξ of m-point spherical n-designs X on S ² is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X ∈ Ξ the minimum spherical distance between points is bounded from below by .) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n ²), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S ². Dedicated to Edward B. Saff on the occasion of his 60th birthday.     

Construction of spherical t-designs

Spherical t-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere S ^d–1, that are exact for polynomials of degree at most t. The concept of such designs was introduced by Delsarte, Goethals and Seidel in 1977. The existence of spherical t-designs for every t and d was proved by Seymour and Zaslavsky in 1984. Although some sporadic examples are known, no general construction has been given. In this paper we give an explicit construction of spherical t-designs on S ^d–1 containing N points, for every t,d and N,NN ₀, where N ₀ = C(d)t ^{O(d 3}).     

Orthogonal basis for spherical monogenics by step two branching

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space \mathbb R^m{{\mathbb R}^m}. They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on \mathbb R^m{{\mathbb R}^m}. Fix the direct sum \mathbb R^m=\mathbb R^p ?\mathbb R^q{{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}. In this article, we will study the decomposition of the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)} of spherical monogenics of order n under the action of Spin(p) × Spin(q). As a result, we obtain a Spin(p) × Spin(q)-invariant orthonormal basis for M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}. In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space M_n(\mathbb R^m, \mathbb C_m){{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}.     

A new technique of integral representations in ?n

A new technique of integral representations in ℂ ⁿ , which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the ∂-equations on strictly pseudoconvex domains in ℂ ⁿ are obtained. These new formulas are simpler than the classical ones, especially the solutions of the ∂-equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in ℂ ⁿ so that all corresponding formulas are simplified.