Polynomial Interpolation on the Unit Sphere and on the Unit Ball

The problem of interpolation on the unit sphere S ^d by spherical polynomials of degree at most n is shown to be related to the interpolation on the unit ball B ^d by polynomials of degree n. As a consequence several explicit sets of points on S ^d are given for which the interpolation by spherical polynomials has a unique solution. We also discuss interpolation on the unit disc of R ² for which points are located on the circles and each circle has an even number of points. The problem is shown to be related to interpolation on the triangle in a natural way.     

A fascinating polynomial sequence arising from an electrostatics problem on the sphere

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance ρ(d) (d is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar’s problem. Using classical potential theory for the harmonic case, we show that 1+ρ(d) is equal to the largest positive zero of a certain sequence of monic polynomials of degree 2d?1 with integer coefficients which we call Gonchar polynomials. Rather surprisingly, ρ(2)?is the Golden ratio and ρ(4) the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.     

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      5. Numerical integration with polynomial exactness over a spherical cap      Kerstin Hesse  Robert S. Womersley 《Advances in Computational Mathematics》2012,36(3):451-483 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 \mathbbSd\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 \mathbbSd\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.      6. Multipliers of spherical harmonics and energy of measures on the sphere      Kathryn E. Hare  Maria Roginskaya 《Arkiv f?r Matematik》2003,41(2):281-294 We consider the operator,f(Δ) for Δ the Laplacian, on spaces of measures on the sphere inR d , show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for theL2-norm off(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure onR d with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.      7. Error estimates for Hermite interpolation on spheres      J. Levesley  Z. Luo 《Journal of Mathematical Analysis and Applications》2003,281(1):46-61 In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere Sd−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.      8. Construction of spherical t-designs      Bela Bajnok 《Geometriae Dedicata》1992,43(2):167-179 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).      9. A Statistic for Testing the Null Hypothesis of Elliptical Symmetry   总被引：1，自引：0，他引：1   A. Manzotti  Francisco J. Prez  Adolfo J. Quiroz 《Journal of multivariate analysis》2002,81(2):274 We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of d. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.      10. On a generalization of distance sets      Hiroshi Nozaki 《Journal of Combinatorial Theory, Series A》2010,117(7):810-826 A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.      11. On the widths of Sobolev classes      Hongwei Huang  Kunyang Wang 《Journal of Complexity》2011,27(2):201-220 Optimal estimates of Kolmogorov’s n-widths, linear n-widths and Gelfand’s n-widths of the weighted Sobolev classes on the unit sphere Sd are established. Similar results are also established on the unit ball Bd and on the simplex Td.      12. Orthogonal polynomials and cubature formulae on balls,simplices, and spheres      《Journal of Computational and Applied Mathematics》2001,127(1-2):349-368 We report on recent developments on orthogonal polynomials and cubature formulae on the unit ball Bd, the standard simplex Td, and the unit sphere Sd. The main result shows that orthogonal structures and cubature formulae for these three regions are closely related. This provides a way to study the structure of orthogonal polynomials; for example, it allows us to use the theory of h-harmonics to study orthogonal polynomials on Bd and on Td. It also provides a way to construct new cubature formulae on these regions.      13. Directional K- Functionals Claudia Cottin      Claudia Cottin 《Results in Mathematics》1993,24(3-4):211-221 When considering approximation of continuous periodic functions f: R d → R by blending-type approximants which depend on directions ξ1,…,ξν ∈ R d directional moduli of smoothness (1) are appropriate measures of smoothness of /. In this paper, we introduce equivalent directional K- functionals. As an application, we obtain a result on the degree of approximation by certain trigonometric blending functions.      14. Weighted Approximation of Functions on the Unit Sphere      Yuan?XuEmail author 《Constructive Approximation》2004,21(1):1-28 The direct and inverse theorems are established for the best approximation in the weighted Lp space on the unit sphere of Rd+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of Rd.      15. On a spherical code in the space of spherical harmonics      A. V. Bondarenko 《Ukrainian Mathematical Journal》2010,62(6):993-996 We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere S d with the use of spaces of spherical harmonics.      16. Riesz External Field Problems on the Hypersphere and Optimal Point Separation      Johann S. Brauchart  Peter D. Dragnev  Edward B. Saff 《Potential Analysis》2014,41(3):647-678 We consider the minimal energy problem on the unit sphere ?? d in the Euclidean space ? d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.      17. Optimal Approximation on Sd      《Journal of Complexity》2000,16(2):424-458 The asymptotic behavior of the n-widths of a wide range of sets of smooth functions on a d-dimensional sphere in Lq(Sd) is studied. Upper and lower bounds for the n-widths are established. Moreover, it is shown that these upper and lower bounds coincide for some important concrete examples.      18. A boundary parametric approximation to the linearized scalar potential magnetostatic field problem      James H. Bramble  Joseph E. Pasciak 《Applied Numerical Mathematics》1985,1(6):493-514 We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary conditions can also be satisfied in the parametric framework. That is the field in the exterior of a sphere is expanded in a ‘harmonic series’ of eigenfunctions for the exterior harmonic problem. The approach is essentially a finite element method coupled with a spectral method via a boundary parametric procedure. The reformulated problem is discretized by finite element techniques which leads to a discrete parametric problem which can be solved by well conditioned iteration involving only the solution of decoupled Neumann type elliptic finite element systems and L2 projection onto subspaces of spherical harmonics. Error and stability estimates given show exponential convergence in the degree of the spherical harmonics and optimal order convergence with respect to the finite element approximation for the resulting fields in L2.      19. Relaxation in SBV p ?(��; S d�C1)      Gra?a Carita  Irene Fonseca  Giovanni Leoni 《Calculus of Variations and Partial Differential Equations》2011,42(1-2):211-255 An integral representation formula is obtained for the relaxation of a class of energy functionals defined in the class of SBV p functions that are constrained to have values on the sphere S d?C1.      20. The capability of approximation for neural networks interpolant on the sphere      Feilong Cao  Shaobo Lin 《Mathematical Methods in the Applied Sciences》2011,34(4):469-478 Compared with planar hyperplane, fitting data on the sphere has been an important and active issue in geoscience, metrology, brain imaging, and so on. In this paper, using a functional approach, we rigorously prove that for given distinct samples on the unit sphere there exists a feed‐forward neural network with single hidden layer which can interpolate the samples, and simultaneously near best approximate the target function in continuous function space. Also, by using the relation between spherical positive definite radial basis functions and the basis function on the Euclidean space ?d + 1, a similar result in a spherical Sobolev space is established. Copyright © 2010 John Wiley & Sons, Ltd.

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.     

Multipliers of spherical harmonics and energy of measures on the sphere

We consider the operator,f(Δ) for Δ the Laplacian, on spaces of measures on the sphere inR ^d , show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for theL²-norm off(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure onR ^d with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.     

Error estimates for Hermite interpolation on spheres

In this paper, we prove convergence rates for spherical spline Hermite interpolation on the sphere S^d−1 via an error estimate given in a technical report by Luo and Levesley. The functionals in the Hermite interpolation are either point evaluations of pseudodifferential operators or rotational differential operators, the desirable feature of these operators being that they map polynomials to polynomials. Convergence rates for certain derivatives are given in terms of maximum point separation.     

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}).     

A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of ^d. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.     

On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.     

On the widths of Sobolev classes

Optimal estimates of Kolmogorov’s n-widths, linear n-widths and Gelfand’s n-widths of the weighted Sobolev classes on the unit sphere S^d are established. Similar results are also established on the unit ball B^d and on the simplex T^d.     

Orthogonal polynomials and cubature formulae on balls,simplices, and spheres

We report on recent developments on orthogonal polynomials and cubature formulae on the unit ball B^d, the standard simplex T^d, and the unit sphere S^d. The main result shows that orthogonal structures and cubature formulae for these three regions are closely related. This provides a way to study the structure of orthogonal polynomials; for example, it allows us to use the theory of h-harmonics to study orthogonal polynomials on B^d and on T^d. It also provides a way to construct new cubature formulae on these regions.     

Directional K- Functionals Claudia Cottin

When considering approximation of continuous periodic functions f: R ^d → R by blending-type approximants which depend on directions ξ₁,…,ξ_ν ∈ R ^d directional moduli of smoothness (1) are appropriate measures of smoothness of /. In this paper, we introduce equivalent directional K- functionals. As an application, we obtain a result on the degree of approximation by certain trigonometric blending functions.     

Weighted Approximation of Functions on the Unit Sphere

The direct and inverse theorems are established for the best approximation in the weighted L^p space on the unit sphere of R^d+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of R^d.     

On a spherical code in the space of spherical harmonics

We propose a new method for the construction of new “nice” configurations of vectors on the unit sphere S ^d with the use of spaces of spherical harmonics.     

Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere ?? ^d in the Euclidean space ? ^d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r ^s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? ^d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? ^d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.     

Optimal Approximation on Sd

The asymptotic behavior of the n-widths of a wide range of sets of smooth functions on a d-dimensional sphere in L_q(S^d) is studied. Upper and lower bounds for the n-widths are established. Moreover, it is shown that these upper and lower bounds coincide for some important concrete examples.     

A boundary parametric approximation to the linearized scalar potential magnetostatic field problem

We consider the linearized scalar potential formulation of the magnetostatic field problem in this paper. Our approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary conditions can also be satisfied in the parametric framework. That is the field in the exterior of a sphere is expanded in a ‘harmonic series’ of eigenfunctions for the exterior harmonic problem. The approach is essentially a finite element method coupled with a spectral method via a boundary parametric procedure. The reformulated problem is discretized by finite element techniques which leads to a discrete parametric problem which can be solved by well conditioned iteration involving only the solution of decoupled Neumann type elliptic finite element systems and L² projection onto subspaces of spherical harmonics. Error and stability estimates given show exponential convergence in the degree of the spherical harmonics and optimal order convergence with respect to the finite element approximation for the resulting fields in L².     

Relaxation in SBV p ?(��; S d�C1)

An integral representation formula is obtained for the relaxation of a class of energy functionals defined in the class of SBV _p functions that are constrained to have values on the sphere S ^d?C1.     

The capability of approximation for neural networks interpolant on the sphere

Compared with planar hyperplane, fitting data on the sphere has been an important and active issue in geoscience, metrology, brain imaging, and so on. In this paper, using a functional approach, we rigorously prove that for given distinct samples on the unit sphere there exists a feed‐forward neural network with single hidden layer which can interpolate the samples, and simultaneously near best approximate the target function in continuous function space. Also, by using the relation between spherical positive definite radial basis functions and the basis function on the Euclidean space ?^{d + 1}, a similar result in a spherical Sobolev space is established. Copyright © 2010 John Wiley & Sons, Ltd.