Spherical spline interpolation—basic theory and computational aspects

The purpose of the paper is to adapt to the spherical case the basic theory and the computational method known from surface spline interpolation in Euclidean spaces. Spline functions are defined on the sphere. The solution process is made simple and efficient for numerical computation. In addition, the convergence of the solution obtained by spherical spline interpolation is developed using estimates for Legendre polynomials.     

Spherical harmonic analysis on affine buildings

Let be a locally finite regular affine building with root system R. There is a commutative algebra spanned by averaging operators A _λ , λ ∈ P ⁺, acting on the space of all functions f:V _P → , where V _P is in most cases the set of all special vertices of , and P ⁺ is a set of dominant coweights of R. This algebra is studied in [6] and [7] for à _n buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h: may be expressed in terms of the Macdonald spherical functions. We also provide a second formula for these homomorphisms in terms of an integral over the boundary of . We may regard as a subalgebra of the C ^*-algebra of bounded linear operators on ?²(V _P ), and we write for the closure of in this algebra. We study the Gelfand map , where M ₂= , and we compute M ₂ and the Plancherel measure of . We also compute the ?²-operator norms of the operators A _λ , λ ∈ P ⁺, in terms of the Macdonald spherical functions.     

球面神经网络的构造与逼近

研究球面神经网络的构造与逼近问题.利用球面广义的de la Vallee Poussin平均、球面求积公式及改进的单变量Cardaliaguet-Euvrard神经网络算子,构造具logistic激活函数的单隐层前向网络,并给出了Jackson型误差估计.     

Stability of Spherical Hermite Interpolation Using Radial Basis Function

In this paper, we give a lower bound for the least eigenvalue of theinterpolation matrix arising from Hermite interpolation by positive definitekernel on sphere , determining the main factor for stabilityof interpolation.     

Spherical Orthotomic and Spherical Antiothotomic

In this paper the spherical orthotomic and the spherical antiorthotomic are defined and their local diffeomorphic types are determined.     

On a Strong Solution in the Method of Spherical Harmonics for a Nonstationary Transport Equation

We prove existence of a strong generalized solution in the Sobolev space W ₂ ¹ to the nonstationary problem for the system of the method of spherical harmonics (MSH) corresponding to the radiation transport problem.     

Mixed spectral method for three-dimensional exterior problems using spherical harmonic and generalized Laguerre functions

In this paper, we develop the mixed spectral method for three-dimensional exterior problems, using spherical harmonic and generalized Laguerre functions. Some basic approximation results are established. The mixed spectral schemes are proposed for two model problems. Their convergences are proved. Numerical results demonstrate the efficiency of this new approach.     

Aliasing Effects and Sampling Theorems of Spherical Random Fields when Sampled on a Finite Grid

Aliasing effects are investigated for spherical random fields sampled on a finite grid. Using the spherical harmonics expansion, it is shown that for a band-limited spherical random field its trend and spectrum can be uniquely reconstructed from the sampled field if the sampling points are judiciously designed. Analytical expressions are also obtained for aliasing errors in the trend and the spectrum when the field is not band-limited.     

On Riesz systems of harmonic conjugates in

In continuation of recent studies, we discuss two constructive approaches for the generation of harmonic conjugates to find null solutions to the Riesz system in . This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. Our first algorithm for harmonic conjugates is based on special systems of homogeneous harmonic and monogenic polynomials, whereas the second one is presented by means of an integral representation. Some examples of function spaces illustrating the techniques involved are given. More specifically, we discuss the (monogenic) Hardy and weighted Bergman spaces on the unit ball in consisting of functions with values in the reduced quaternions. We end up proving the boundedness of the underlying harmonic conjugation operators in certain weighted spaces. Copyright © 2013 John Wiley & Sons, Ltd.     

n次1阶球谐函数的加法公式及其应用

利用坐标旋转后球坐标变量间的关系和连带勒让德函数的性质导出了n次1阶球谐函数的加法公式,并以一个实际的电磁场边值问题为例介绍了n次1阶球谐函数加法公式的应用.     

Eigenvalues of functions of orthogonal projectors

By representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices, several characterizations concerning eigenvalues of various functions of the pair are obtained. These results substantially extend the ones already available in the literature. Additionally, some related results dealing with the functions of a pair of orthogonal projectors are provided, with the emphasis laying on the problem of invertibility.     

Spherical and Clockwise Spherical Graphs

The main subject of our study are spherical (weakly spherical) graphs, i.e. connected graphs fulfilling the condition that in each interval to each vertex there is exactly one (at least one, respectively) antipodal vertex. Our analysis concerns properties of these graphs especially in connection with convexity and also with hypercube graphs. We deal e.g. with the problem under what conditions all intervals of a spherical graph induce hypercubes and find a new characterization of hypercubes: G is a hypercube if and only if G is spherical and bipartite.     

Experiments on orthogonalization by biorthogonal representations of orthogonal projectors

A number of experiments are performed with the aim of enhancing a particular feature arising when biorthogonal sequences are used for the purpose of orthogonalization. It is shown that an orthogonalization process executed by biorthogonal sequences and followed by a re-orthogonalization step admits four numerically different realizations. The four possibilities are originated by the fact that, although an orthogonal projector is by definition a self-adjoint operator, due to numerical errors in finite precision arithmetic the biorthogonal representation does not fulfil such a property. In the experiments presented here one of the realizations is shown clearly numerically superior to the remaining three.     

球壳中球形夹杂对SH波的三维散射与动应力集中

研究了一般情况下球壳中球形夹杂(包括孔洞)引起的SH波三维散射与动应力集中现象.根据球壳与夹杂的几何特点,分别以球壳和夹杂中心建立球坐标,用于描述球壳中的入射波、散射波和夹杂中的驻波势函数,并采用球波函数的加法公式,实现了不同坐标下球波函数的变换,推导出位移、应力分量的解析解.结合球壳的边界条件和夹杂界面的连续条件,求解了不同材料属性夹杂,以及空洞情况下弹性波的散射和动应力集中因子分布情况,并分析了频率以及夹杂中心位置对动应力集中因子的影响.文中的研究为球壳结构的力学性能分析以及无损检测提供了理论支持.     

圆底扁球壳非线性稳定问题的自由参数摄动法解

用自由参数摄动法求解了圆底扁球壳在均布载荷作用下的非线性稳定问题．作为一种改进的正则摄动方法,使研究者可以不确定摄动参数具体意义而直接求得问题的全部特征方程．通过算例研究了扁球壳在失稳过程中变形和应力的变化特点,并与其他研究者的结果进行了比较．     

Complex Spherical Designs and Codes

Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper, we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these association schemes are nonsymmetric.     

Spherical unitary highest weight representations

In this paper we give an almost complete classification of the -spherical unitary highest weight representations of a hermitian Lie group , where is a symmetric space of Cayley type.

     

Spherical Minimal Immersions with Prescribed Codimension

We describe a general construction of manufacturing new spherical minimal immersions between round spheres out of old ones. The new immersions have higher domain dimension and degree and the construction has a precise control on the codimension. Applied to classified and recent examples, the construction gives an abundance of new spherical minimal immersions with prescribed codimensions.Mathematics Subject Classifications (2000). 53C42     

入射平面电磁波两类球面波函数展开式的等价性证明

入射平面电磁波的球面波函数展开是求解不同圆球结构的平面波散射问题的重要工具,相关文献分别利用场的坐标分解和矢量势法得到了入射平面波的球面波函数的两种不同形式的展开式.利用偏微分方程边值问题解的存在唯一性定理,给出了这两种展开式的等价性的一个简洁的解析证明,并进行了数值验证.