共查询到17条相似文献,搜索用时 93 毫秒
1.
借助于球调和多项式的de la Vallée Poussin和构造出了单位球面S^q上一类带形平移网络算子,并给出了其对L^p(S^q)中函数一致逼近的收敛速度.
相似文献
2.
构造了具有插值性质的球面带形平移网络, 并且给出了在一致范数下对连续函数逼近的上界估计. 相似文献
3.
借助于经典球面分析的Bochner-Riesz平均,Cesàro平均及有关球调和多项式的Gauss积分公式构造出了两类球面平移算子,并且以K-泛函为工具给出了逼近的上界估计. 相似文献
4.
研究了球型平移网络对周期函数的逼近问题.文章首先将基函数eimx分别表示成为两种球型平移网络.进一步,将有关多重Fourier级数的Bochner-Riesz平均表示成为球型平移网络的形式.在此基础上构造出了两类球型平移网络序列,并借助于有关Bochner-Riesz平均对Lp空间中函数的逼近结果给出了这两类球型平移网络序列在Lp空间中的逼近阶. 相似文献
5.
提出了一种用于两同心球所介球形区域上流体低马赫数流动的全离散混合Legendre-球面调和谱格式,即时间方向上一阶向前差商代替时间方向导数,半径方向用Legendre正交逼近,球面方向上用球面调和正交逼近.严格证明了格式在不同参数组合下的广义稳定性,并通过数值结果显示了格式的高精度. 相似文献
6.
本文对于单位球面上的经典连续模,给出了一个非常有用的广义Ul'yanov型不等式.该不等式在球面多项式逼近、球面嵌入理论以及球面上函数空间的插值理论等领域有着非常重要的应用.我们的证明基于球面调和多项式展开的新的估计,这些估计本身也具有独立的意义. 相似文献
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主要研究两同心球所界球形区域上偏微分方程的谱方法,建立了与区域形状相适应的混合Legendre-球面调和正交逼近的部分结果,在此基础上提出了数值求解两同心球所界球形区域上Fisher型方程的混合Legendre-球面调和谱格式,并分别给出了格式的收敛性及相关的数值结果. 相似文献
11.
This paper describes a new approach to the problem of computing spherical expansions of zonal functions on Euclidean spheres.
We derive an explicit formula for the coefficients of the expansion expressing them in terms of the Taylor coefficients of
the profile function rather than (as done usually) in terms of its integrals against Gegenbauer polynomials. Our proof of
this result is based on a polynomial identity equivalent to the canonical decomposition of homogeneous polynomials and uses
only basic properties of this decomposition together with simple facts concerning zonal harmonic polynomials. As corollaries,
we obtain direct and apparently new derivations of the so-called plane wave expansion and of the expansion of the Poisson
kernel for the unit ball.
Received: 26 January 2007 相似文献
12.
We introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball. 相似文献
13.
A. Moiola R. Hiptmair I. Perugia 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,62(5):809
In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω 2 u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size, and the number of plane waves used in the approximations. 相似文献
14.
Plane wave approximation of homogeneous Helmholtz solutions 总被引:1,自引:0,他引:1
A. Moiola R. Hiptmair I. Perugia 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2011,14(4):809-837
In this paper, we study the approximation of solutions of the homogeneous Helmholtz equation Δu + ω
2
u = 0 by linear combinations of plane waves with different directions. We combine approximation estimates for homogeneous Helmholtz
solutions by generalized harmonic polynomials, obtained from Vekua’s theory, with estimates for the approximation of generalized
harmonic polynomials by plane waves. The latter is the focus of this paper. We establish best approximation error estimates
in Sobolev norms, which are explicit in terms of the degree of the generalized polynomial to be approximated, the domain size,
and the number of plane waves used in the approximations. 相似文献
15.
J.M. Melenk 《Numerische Mathematik》1999,84(1):35-69
Summary. The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted
systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation
properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rates
of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is
paid to the approximation of singular functions that arise typically in corners. These results for harmonic polynomials are
extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximation
results for Laplace's equation hold true verbatim, if harmonic polynomials are replaced with generalized harmonic polynomials.
The Partition of Unity Method is used in a numerical example to construct an operator adapted spectral method for Laplace's
equation that is based on approximating with harmonic polynomials locally.
Received May 26, 1997 / Revised version received September 21, 1998 / Published online September 7, 1999 相似文献
16.
PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units 下载免费PDF全文
Deep neural network with rectified linear units (ReLU) is getting more and
more popular recently. However, the derivatives of the function represented by a ReLU
network are not continuous, which limit the usage of ReLU network to situations only
when smoothness is not required. In this paper, we construct deep neural networks
with rectified power units (RePU), which can give better approximations for smooth
functions. Optimal algorithms are proposed to explicitly build neural networks with
sparsely connected RePUs, which we call PowerNets, to represent polynomials with
no approximation error. For general smooth functions, we first project the function to
their polynomial approximations, then use the proposed algorithms to construct corresponding PowerNets. Thus, the error of best polynomial approximation provides an
upper bound of the best RePU network approximation error. For smooth functions in
higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product
grid and sparse grid discretization to get polynomial approximations. Our constructive algorithms show clearly a close connection between spectral methods and deep
neural networks: PowerNets with $n$ hidden layers can exactly represent polynomials
up to degree $s^n$, where $s$ is the power of RePUs. The proposed PowerNets have potential applications in the situations where high-accuracy is desired or smoothness is
required. 相似文献
17.
The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli
of smoothness. The treatment applies to a wide class of Banach spaces of functions.
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