一种基于多样化多项式的高概率稀疏插值算法

稀疏多元多项式插值用于构造黑盒函数,是求解多项式代数问题的一种有效策略,具有多项式时间复杂度的多元稀疏插值算法已得到广泛研究和使用.近期Huang(2021)提出了一个基于多样化多项式的稀疏插值算法,计算复杂度为O(nTlog2q+nT√Dlogq),是有限域上首个关于变元个数n和项数界T的线性函数,关于次数界D的分数...     

S.N.Berns型三角插值多项式

1．引言由Faber定理［1］可知，以任何点组作为插值节点的函数g（t）的Lagrange三角插值多项式算子并非对每个连续的周期函数都能在全实轴上一致地收敛．为改善其收敛性，Bernstein在［2］中将Lagrange插值基函数作平均，得算子Zn—1其中为插值节点，为ragrange三角插值多项式的基函数．O．K。。在1969年t3］得到估计式／43＼／7T＼ig（t）一on（g，t）l三卜十三）w（；“）．＼7TZ八Th／他于1973年［4J将上面的估计式改进为19／7T＼ig（t）一Cn（g，t）155叫g，“）．“””’“””’”一QnV’n／［4］中还引进算子B。（g，t）＝＝…     

新组合型的三角插值多项式

将被插函数进行组合平均,构造一个新组合型的三角插值多项式Cn(f;t,x),使得它在全轴上一致收敛到每个以2π为周期的连续函数,且对Cj2π连续函数类的逼近阶达到最佳,这里0jt,t为任给的奇自然数.     

基于第二类Chebyshev多项式零点的Pal-型插值多项式的逼近

设{x_k}_(k-0)~n是n 1次多项式U_n(x)=(1-x~2)U_n(x)的零点,其中U_n(x)是第二类Chebyshev多项式。设是的零点。根据Pal的插值理论,对函数f∈C~1[-1,1],存在唯一的2n 1次多项式满足条件: 本文研究用Pal型插值多项式对函数f∈C~r[-1,1](r≥1)和它的导函数的逼近。     

矩阵的Kronecker积的性质

利用矩阵初等变换的结果,给出矩阵的Kronecker积的Jordan标准形,进而得到矩阵的Kronecker积的最小多项式和可相似对角化的条件,最后归纳总结了矩阵的Kronecker积的范数和分解的结果.     

Bergman空间的插值多项式逼近

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基于Chebyshev多项式零点的Lagrange插值多项式逼近的注记

本文通过一个例子说明了文献[3]中定理6.9的不完善之处,并建立了:若f∈C^r_[-1,1],则     

多项式插值余项定理的一个自然证明

通过对插值多项式函数性质进行分析,多项式插值余项的基本形式得到诱导,再从该基本形式出发,获得了多项式插值余项定理的新证明.整个证明过程无需借助辅助函数的构造,因而显得较为自然.这种自然证明的方式也可用于Hermite切触型插值多项式余项的证明.     

Grunwald插值多项式算子的逼近阶

设J_n~(α,β)(x)(α,β>-1)是在[-1,1]上以ρ(x)=(1-x)~α(1+x)~β为权函数的n阶Jacobi正交多项式。l_k~(n)(x)(K=1,2,…,n)是以J_n~(α,β)(x)的零点{x~(n)_1,x_2~(n),…,X_n~(n)}为基点的Lagrange插值基本多项式,对于f(x)∈C[-1,1],其Grunwald插值多项式算子是(见[1]第Ⅲ部分;[2]P.196)     

三角多项式类中的Birkhoff型三角插值

本文在三角多项式类中讨论了2π周期函数的一类Birkhoff型等距结点的三角插值问题,给出了此问题有解的充要条件,并构造出插值基.     

On interpolation by radial polynomials

A lemma of Micchelli's, concerning radial polynomials and weighted sums of point evaluations, is shown to hold for arbitrary linear functionals, as is Schaback's more recent extension of this lemma and Schaback's result concerning interpolation by radial polynomials. Schaback's interpolant is explored. Happy 60th and beyond, Charlie! Mathematics subject classifications (2000) 41A05, 41A6.     

Functions approximated by any sequence of interpolation polynomials

In this note, we seek for functions f which are approximated by the sequence of interpolation polynomials of f obtained by any prescribed system of nodes.     

High dimensional polynomial interpolation on sparse grids

We study polynomial interpolation on a d-dimensional cube, where d is large. We suggest to use the least solution at sparse grids with the extrema of the Chebyshev polynomials. The polynomial exactness of this method is almost optimal. Our error bounds show that the method is universal, i.e., almost optimal for many different function spaces. We report on numerical experiments for d = 10 using up to 652 065 interpolation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Deterministic sampling of sparse trigonometric polynomials

One can recover sparse multivariate trigonometric polynomials from a few randomly taken samples with high probability (as shown by Kunis and Rauhut). We give a deterministic sampling of multivariate trigonometric polynomials inspired by Weil’s exponential sum. Our sampling can produce a deterministic matrix satisfying the statistical restricted isometry property, and also nearly optimal Grassmannian frames. We show that one can exactly reconstruct every M-sparse multivariate trigonometric polynomial with fixed degree and of length D from the determinant sampling X, using the orthogonal matching pursuit, and with |X| a prime number greater than (MlogD)². This result is optimal within the (logD)² factor. The simulations show that the deterministic sampling can offer reconstruction performance similar to the random sampling.     

Critical pairs of matrices for the degree of the minimal polynomial of the Kronecker sum

In [4] Dias da Silva and Hamidoune obtained a lower bound for the degree of the minimal polynomial of the Kronecker sum of two matrices in terms of the degrees of the minimal polynomials of the matrices. We determine the pairs of matrices for which this bound is reached.     

Periodic interpolation and wavelets on sparse grids

Nested spaces of multivariate periodic functions forming a non-stationary multiresolution analysis are investigated. The scaling functions of these spaces are fundamental polynomials of Lagrange interpolation on a sparse grid. The approach based on Boolean sums leads to sample and wavelet spaces of significantly lower dimension and good approximation order. The algorithms for complete decomposition and reconstruction are of simple structure and low complexity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solutions to a family of matrix equations by using the Kronecker matrix polynomials

Closed form solutions to a family of generalized Sylvester matrix equation in form of are given by using the so-called Kronecker matrix polynomials. It is found that the structure of the solutions is independent of the orders ?,ψ and φ. This type of uniform closed form solutions includes our early results as special cases. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in linear systems.     

Pseudozeros of multivariate polynomials

The pseudozero set of a system of polynomials in complex variables is the subset of which is the union of the zero-sets of all polynomial systems that are near to in a suitable sense. This concept is made precise, and general properties of pseudozero sets are established. In particular it is shown that in many cases of natural interest, the pseudozero set is a semialgebraic set. Also, estimates are given for the size of the projections of pseudozero sets in coordinate directions. Several examples are presented illustrating some of the general theory developed here. Finally, algorithmic ideas are proposed for solving multivariate polynomials.

     

Approximate decoupling of multivariate polynomials using weighted tensor decomposition

Many scientific and engineering disciplines use multivariate polynomials. Decomposing a multivariate polynomial vector function into a sandwiched structure of univariate polynomials surrounded by linear transformations can provide useful insight into the function while reducing the number of parameters. Such a decoupled representation can be realized with techniques based on tensor decomposition methods, but these techniques have only been studied in the exact case. Generalizing the existing techniques to the noisy case is an important next step for the decoupling problem. For this extension, we have considered a weight factor during the tensor decomposition process, leading to an alternating weighted least squares scheme. In addition, we applied the proposed weighted decoupling algorithm in the area of system identification, and we observed smaller model errors with the weighted decoupling algorithm than those with the unweighted decoupling algorithm.     

On some aspects of multivariate polynomial interpolation

The purpose of this paper is to present some aspects of multivariate Hermite polynomial interpolation. We do not focus on algebraic considerations, combinatoric and geometric aspects, but on explicitation of formulas for uniform and non-uniform bivariate interpolation and some higher dimensional problems. The concepts of similar and equivalent interpolation schemes are introduced and some differential aspects related to them are also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.