Numerical properties of generalized discrepancies on spheres of arbitrary dimension

Quantifying uniformity of a configuration of points on the sphere is an interesting topic that is receiving growing attention in numerical analysis. An elegant solution has been provided by Cui and Freeden [J. Cui, W. Freeden, Equidistribution on the sphere, SIAM J. Sci. Comput. 18 (2) (1997) 595–609], where a class of discrepancies, called generalized discrepancies   and originally associated with pseudodifferential operators on the unit sphere in R³${\mathbb{R}}^3$, has been introduced. The objective of this paper is to extend to the sphere of arbitrary dimension this class of discrepancies and to study their numerical properties. First we show that generalized discrepancies are diaphonies on the hypersphere. This allows us to completely characterize the sequences of points for which convergence to zero of these discrepancies takes place. Then we discuss the worst-case error of quadrature rules and we derive a result on tractability of multivariate integration on the hypersphere. At last we provide several versions of Koksma–Hlawka type inequalities for integration of functions defined on the sphere.     

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.     

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}).     

Clifford Analysis Techniques for Spherical PDE

For a ? R\alpha \in \mathbf{R}, the class of a-\alpha -order spherical harmonic functions in an open set W í\Omega \subseteq S^n-1\mathbf{S}^{n-1}, H^a(W)H^{\alpha }(\Omega ) is defined as the C²-C^{2}-solutions of D_au=0\Delta _{\alpha }u=0; where D_a=D_s+a(n+a-2)\Delta _{\alpha }=\Delta _{s}+\alpha (n+\alpha -2) is the spherical Laplace--Beltrami operator of order a\alpha and D_s\Delta _{s} is the radially independent part of the Laplace operator. We obtain a Green's integral formula for the functions in H^a(W)H^{\alpha }(\Omega ) with kernel expressed as a Gegenbauer function. As generalizations, higher order spherical iterated Dirac operators are defined in a polynomial form. Integral representations of the null solutions to these operators and an intertwining formula relating these operators on the sphere and their analogues in Euclidean space are presented.     

Quasi–Monte Carlo rules for numerical integration over the unit sphere {\mathbb{S}^2}

We study numerical integration on the unit sphere ${\mathbb{S}^2 \subseteq\mathbb{R}^3}$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a (0, m, 2)-net given in the unit square [0, 1]² to the sphere ${\mathbb{S}^2}$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J Sci Comput 18(2):595–609, 1997]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on ${\mathbb{S}^2}$ . And finally, we prove an upper bound on the spherical cap L ₂-discrepancy of order N ^?1/2(log N)^1/2 (where N denotes the number of points). This improves upon the bound on the spherical cap L ₂-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Commun Pure Appl Math 39(S, suppl):S149–S186, 1986] by a factor of ${\sqrt{\log N}}$ . Numerical results suggest that the (0, m, 2)-nets lifted to the sphere ${\mathbb{S}^2}$ have spherical cap L ₂-discrepancy converging with the optimal order of N ^?3/4.     

Separation of the elasticity theory equations with radial inhomogeneity : PMM vol. 38, n 6, 1974, pp. 1139–1144

The separation of a system of three elasticity theory equations in the static case to a system of two equations and one independent equation for a space with a radial inhomogeneity is presented in a spherical coordinate system. These equations are solved by separation of variables for specific kinds of radial inhomogeneity. In particular, solutions are found for the Lamé coefficients μ = const, λ (ifr) is an arbitrary function, μ = μ_or^β, λ = λ_or^β.While methods of solving problems associated with the equilibrium of an elastic homogeneous sphere have been studied sufficiently [1], problems with spherical symmetry of the boundary conditions have mainly been solved for an inhomogeneous sphere [2, 3],For a particular kind of inhomogeneity dependent on one Cartesian coordinate, the equations have been separated completely in [4], A system of three equations with a radial inhomogeneity in a spherical coordinate system is separated below by a method analogous to [4].     

Spherical 7-Designs in 2 n -Dimensional Euclidean Space

We consider a finite subgroup _n of the group O(N) of orthogonal matrices, where N = 2 ⁿ , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 ⁿ -dimensional Euclidean space R ^N . We prove that representations of the group _n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit _n,2 x ^t of any initial point x on the sphere S _{N – 1} is a 7-design in the Euclidean space of dimension 2 ⁿ .     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

A Point Balance Algorithm for the Spherical Code Problem

The Spherical Code (SC) problem has many important applications in such fields as physics, molecular biology, signal transmission, chemistry, engineering and mathematics. This paper presents a bilevel optimization formulation of the SC problem. Based on this formulation, the concept of balanced spherical code is introduced and a new approach, the Point Balance Algorithm (PBA), is presented to search for a 1-balanced spherical code. Since an optimal solution of the SC problem (an extremal spherical code) must be a 1-balanced spherical code, PBA can be applied easily to search for an extremal spherical code. In addition, given a certain criterion, PBA can generate efficiently an approximate optimal spherical code on a sphere in the n-dimensional space ⁿ. Some implementation issues of PBA are discussed and putative global optimal solutions of the Fekete problem in 3, 4 and 5-dimensional space are also reported. Finally, an open question about the geometry of Fekete points on the unit sphere in the 3-dimensional space is posed.     

Entropy and Convergence on Compact Groups

We investigate the behaviour of the entropy of convolutions of independent random variables on compact groups. We provide an explicit exponential bound on the rate of convergence of entropy to its maximum. Equivalently, this proves convergence of the density to uniformity, in the sense of Kullback–Leibler. We prove that this convergence lies strictly between uniform convergence of densities (as investigated by Shlosman and Major), and weak convergence (the sense of the classical Ito–Kawada theorem). In fact it lies between convergence in L ¹⁺ and convergence in L ¹.     

Solution of one problem of nonlinear adsorption kinetics by the method of lines

The method of lines is constructed and proved for numerical solution of a nonlinear initial-boundary-value problem of parabolic type describing the adsorption of a substance from an aqueous solution of bounded volume by a spherical adsorbent. The method is developed under natural assumptions on the smoothness of the solution of the original problem. The rate of convergence of the method depends on the smoothness of the initial function and is of order O(h) if v₀(x) 0, O(h^1/2) if v₀(x) C¹[0, 1], and 0(|v ₀(x)|W ₂ ¹ (O,h)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 24–30, 1987.     

An interior point multiplicative method for optimization under positivity constraints

We analyze an algorithm for the problem minf(x) s.t.x 0 suggested, without convergence proof, by Eggermont. The iterative step is given by x _j ^k+1 =x _j ^k (1-_kf(x ^k)_j) with _k > 0 determined through a line search. This method can be seen as a natural extension of the steepest descent method for unconstrained optimization, and we establish convergence properties similar to those known for steepest descent, namely weak convergence to a KKT point for a generalf, weak convergence to a solution for convexf and full convergence to the solution for strictly convexf. Applying this method to a maximum likelihood estimation problem, we obtain an additively overrelaxed version of the EM Algorithm. We extend the full convergence results known for EM to this overrelaxed version by establishing local Fejér monotonicity to the solution set.Research for this paper was partially supported by CNPq grant No 301280/86.     

The convergence of multidimensional fourier-bessel series

We study the convergence of series of eigenfunctions of the Laplacian in the unit ballB ^d. The problem is posed in the spacesL _rad ^p (L _ang ² ). A convergence result is obtained in the sharp range2d/(d + 1) <p <2d/(d-1). There is a close connection with the spherical summation of classical trigonometric expansions. The proofs involve weighted inequalities for singular integrals, as well as a precise decomposition of oscillatory integrals using van der Corput’s method.     

Convergence of intermediate rows of minimal polynomial and reduced rank extrapolation tables

Let {x _m} _m ^=0/ be a vector sequence obtained from a linear fixed point iterative technique in a general inner product space. In two previous papers [6,9] the convergence properties of the minimal polynomial and reduced rank extrapolation methods, as they are applied to the vector sequence above, were analyzed. In particular, asymptotically optimal convergence results pertaining to some of the rows of the tables associated with these two methods were obtained. In the present work we continue this analysis and provide analogous results for the remaining (intermediate) rows of these tables. In particular, when {x _m} _m ^=0/ is a convergent sequence, the main result of this paper says, roughly speaking, that all of the rows converge, and it also gives the rate of convergence for each row. The results are demonstrated numerically through an example.     

Spectral approximation in the numerical stability study of nondivergent viscous flows on a sphere

The accuracy of calculating the normal modes in the numerical linear stability study of two-dimensional nondivergent viscous flows on a rotating sphere is analyzed. Discrete spectral problems are obtained by truncating Fourier's series of the spherical harmonics for both the basic flow and the disturbances to spherical polynomials of degrees K and N, respectively. The spectral theory for the closed operators [1], and embedding theorems for the Hilbert and Banach spaces of smooth functions on a sphere are used to estimate the rate of convergence of the eigenvalues and eigenvectors. It is shown that the convergence takes place if the basic state is sufficiently smooth, and the truncation numbers K and N of Fourier's series for the basic flow and disturbances tend to infinity keeping the ratio N/K fixed. The convergence rate increases with the smoothness of the basic flow and with the power s of the Laplace operator in the vorticity equation diffusion term. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14:143–157, 1998     

Spherical f-Tilings by (Equilateral and Isosceles) Triangles and Isosceles Trapezoids

Dihedral f-tilings by spherical parallelograms and spherical triangles were obtained in [3–5]. In this paper we extend these results presenting the study of all dihedral f-tilings of the sphere S ², whose prototiles are a spherical equilateral or isosceles triangle and a spherical isosceles trapezoid. The combinatorial structure, including the symmetry group of each tiling, is given in Table 1.     

Spherical wavelet transform and its discretization

A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to wavelet packets and scale discrete wavelets. The essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (C ₀) (like Abel-Poisson or Gauß-Weierstraß operators) lead in a canonical way to (pyramidal) algorithms.Supported by the Graduiertenkolleg Technomathematik, Kaiserslautern.     

Carathéodory balls and norm balls in a class of convex bounded Reinhardt domains

A family of the spherical fractional integrals on the unit sphere Σ _n in ℝ ⁿ⁺¹ is investigated. This family includes the spherical Radon transform (α = 0) and the Blaschke-Levy representation (α>1). Explicit inversion formulas and a characterization ofT ^αƒ are obtained for ƒ belonging to the spacesC ^∞,C, L^p and for the case when ƒ is replaced by a finite Borel measure. All admissiblen ≥ 2,α ε ℂ, andp are considered. As a tool we use spherical wavelet transforms associated withT ^α. Wavelet type representations are obtained forT ^α ƒ, ƒ εL ^p, in the case Reα ≤ 0, provided thatT ^α is a linear bounded operator inL ^p. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany).     

（0,1;0）-INTERPOLATION ON INFINITE INTERVAL （-∞, ＋∞）

In this paper, we study the explicit representation and convergence of （0, 1;0）-interpolation on infisite interval, which means to determine a polynomial of degree ≤ 3n - 2 when the function values areprescribed at two set of points namely the zeros of Hn（x） and H′n （x） and the first derivatives at the zerosof H′n（x）.     

A variational characterisation of spherical designs

In this paper we first establish a new variational characterisation of spherical designs: it is shown that a set , where , is a spherical L-design if and only if a certain non-negative quantity A_L,N(X_N) vanishes. By combining this result with a known “sampling theorem” for the sphere, we obtain the main result, which is that if is a stationary point set of A_L,N whose “mesh norm” satisfies h_XN<1/(L+1), then X_N is a spherical L-design. The latter result seems to open a pathway to the elusive problem of proving (for fixed d) the existence of a spherical L-design with a number of points N of order (L+1)^d. A numerical example with d=2 and L=19 suggests that computational minimisation of A_L,N can be a valuable tool for the discovery of new spherical designs for moderate and large values of L.