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.     

Locally Adaptive Wavelet Empirical Bayes Estimation of a Location Parameter

The traditional empirical Bayes (EB) model is considered with the parameter being a location parameter, in the situation when the Bayes estimator has a finite degree of smoothness and, possibly, jump discontinuities at several points. A nonlinear wavelet EB estimator based on wavelets with bounded supports is constructed, and it is shown that a finite number of jump discontinuities in the Bayes estimator do not affect the rate of convergence of the prior risk of the EB estimator to zero. It is also demonstrated that the estimator adjusts to the degree of smoothness of the Bayes estimator, locally, so that outside the neighborhoods of the points of discontinuities, the posterior risk has a high rate of convergence to zero. Hence, the technique suggested in the paper provides estimators which are significantly superior in several respects to those constructed earlier.     

单位球面上的一极值问题

设Ω是R~n中的紧集,1相似文献   

Minimum-Variance Multitaper Spectral Estimation on the Sphere

We develop a method to estimate the power spectrum of a stochastic process on the sphere from data of limited geographical coverage. Our approach can be interpreted either as estimating the global power spectrum of a stationary process when only a portion of the data are available for analysis, or estimating the power spectrum from local data under the assumption that the data are locally stationary in a specified region. Restricting a global function to a spatial subdomain—whether by necessity or by design—is a windowing operation, and an equation like a convolution in the spectral domain relates the expected value of the windowed power spectrum to the underlying global power spectrum and the known power spectrum of the localization window. The best windows for the purpose of localized spectral analysis have their energy concentrated in the region of interest while possessing the smallest effective bandwidth as possible. Solving an optimization problem in the sense of Slepian (1960) yields a family of orthogonal windows of diminishing spatiospectral localization, the best concentrated of which we propose to use to form a weighted multitaper spectrum estimate in the sense of Thomson (1982). Such an estimate is both more representative of the target region and reduces the estimation variance when compared to estimates formed by any single bandlimited window. We describe how the weights applied to the individual spectral estimates in forming the multitaper estimate can be chosen such that the variance of the estimate is minimized.     

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.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions

This article presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotationinvariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from classical tensor product wavelet theory.     

Adaptive Estimation of Distributions Using Exponential Sub-Families

Abstract

An algorithm is presented that, for a large-dimensional exponential family G, finds a lower dimension exponential sub-family of G which contains distributions best fitting groups of identically distributed observations within a set of data. The data are therefore fitted to a family of distributions that has been adaptively chosen as representative of them. The algorithm is implemented in the special case in which G is a logspline family of distributions. An example data set is analyzed using the method.     

C~n中单位球面上的不变g-函数及BMOA

设B是C~n中的单位球,S是C~n中的单位球面。■(f)是定义在S上的不变g-函数．设f∈ BMOA,若存在正测度集ES,使■(f)＜ ∞在E上成立,则■(f)＜ ∞在S上几乎处处成立,同时■(f)∈BMO(S)且存在常数C,使得‖■(f)‖≤C‖f‖.     

Cn中单位球面上的不变g-函数及BMOA

设B是Cn中的单位球,S是Cn中的单位球面,(g)(f)是定义在S上的不变g-函数.设f∈BMOA,若存在正测度集E(∪)S,使(g)(f)＜+∞在E上成立,则(g)(f)＜+∞在S上几乎处处成立,同时(g)(f)∈BMO(S)且存在常数C,使得‖(g)(f)‖*≤C‖f‖*.     

n维单位球面上的Toeplitz符号演算

探讨了C^n中单位球面S上Berezin变换和Toeplitz算子的性质，证明了由{Tφ,φ∈L^∞ （S）}所生成的C^＊-代数中算子T的符号恰好为单位球B上函数T（称为T的Berezin变换）的非切向边界值．此外，本文还得到了经典Toeplitz符号演算的有趣推广．     

A Note on Certain Block Spaces on the Unit Sphere

In this note, we clarify a relation between block spaces and the Hardy space. We obtain Bq^0.v belong to H^1（S^n-1）＋L（ln＋L）^1＋v（s^n-1）,v〉-1,q〉1,Furthermore,if v≥ 0, q 〉 1. we verify that block spaces Rq^0.v（S^n-1）are proper subspaces of H1 （S^n- 1）,     

Adaptive Estimation of FCG Using Nonlinear State-Space Models

Abstract

In this paper, an efficient adaptive nonlinear algorithm for estimation and identification, the so-called adaptive Lainiotis filter (ALF), is applied to the problem of fatigue crack growth (FCG) estimation, identification, and prediction of the final crack (failure). A suitable nonlinear state-space FCG model is introduced for both ALF and extended Kalman filter (EKF). Both algorithms are tested in order to compare their efficiency. Through extensive analysis and simulation, it is demonstrated that the ALF has superior performance both in FCG estimation, as well as in predicting the remaining lifetime to failure. Furthermore, it is shown that the ALF is faster and easier to implement in a parallel/distributed processing mode, and much more robust than the classic EKF.     

Approximation by Means of h-Harmonic Polynomials on the Unit Sphere

The h-harmonics are analogues of the ordinary harmonics, they are orthogonal homogeneous polynomials on the sphere with respect to a weight function that is invariant under a reflection group. Two means of associated orthogonal expansions, the de la Vallée Poussin means and an analog of spherical means, are defined and their approximation behaviors are studied. A weighted modulus of smoothness is defined using the modified spherical means and is proved to be equivalent to a weighted K-modulus defined using the differential-difference h-spherical Laplacian. A Bernstein type inequality for the h-spherical Laplacian is also established.     

单位球面上的等距算子的延拓

本文考虑一般的Banach空间上的等距延拓问题,利用赋范集的概念给出了一些充分条件,使得单位球面间的满等距算子可以延拓为全空间上的线性等距算子。     

单位球中的极小曲面

Let M be a closed surface with positive Gauss curvature minimally immersed in a standard Euclidean unit sphere S~n.In this paper,we choose a local orthonormal frame field on M,under which the shape operators have very convenient form.We also give some applications of this kind of frame field.     

Density Estimation on the Stiefel Manifold

This paper develops the theory of density estimation on the Stiefel manifoldV_{k, m}, whereV_{k, m}is represented by the set ofm×kmatricesXsuch thatX′X=I_k, thek×kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldV_{k, m}play useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofV_{k, m}.     

球面均匀分布的拟合优度检验

证明了基于惯量矩的d维单位球面上样本服从均匀分布的基本特征,得到球面均匀分布协差阵特征根估计的强相合性及渐近多元正态性.提出了检验球面上样本均匀性的渐近卡方统计量,证明了拟合优度检验的相合性并做检验功效的随机模拟.     

单位球面中极小子流形的C∞紧性

本文研究了单位球面中极小子流形的C^∞紧性,并得到两个紧性定理.作为应用,我们证明了存在正数δ(n),如果单位球面中极小子流形的第2基本形式的长度平方小于2/3n+δ(n),则它必须是全测地的或微分同胚于Veronese曲面.     

g-Natural Contact Metrics on Unit Tangent Sphere Bundles

We construct a three-parameter family of contact metric structures on the unit tangent sphere bundle T ₁ M of a Riemannian manifold M and we study some of their special properties related to the Levi-Civita connection. More precisely, we give the necessary and sufficient conditions for a constructed contact metric structure to be K-contact, Sasakian, to satisfy some variational conditions or to define a strongly pseudo-convex CR-structure. The obtained results generalize classical theorems on the standard contact metric structure of T ₁ M.