高维频谱有限函数的外推

在实轴 R~1上的限制.因此若已知 f(t)在 t∈[-T,T]上的值,则 f(t)在[-T,T]外的值就被唯一确定了.作为猜测,J.L.C.Sanz 和 T.S.Huang 于1983年提出了一种外推 f(t)在[-T,T]外的值的方法(见[1]).文献[2]和[3]从不同角度证实了这     

关于解析算子值函数的几个结果

关于复值解析函数Riesz—Dunford积分的Ky Fan定理由[1]推广到算子值解析函数,由此函数论中的很多定理得到了推广.本文的目的在于改进[1]中的结果,得到了较弱条件下的Pick定理,从而推广了[2]中的Julia引理,并简化了其证明过程.     

Mathematica软件快速上手指南（2）——数,函数和函数图象

为了对表达式进行计算 ,我们应当先了解系统关于表达式书写的基本规定 ,包括最基本的各种数的表示 ,系统的最常用的各种函数 ,以及如何书写一般的代数表达式 .1　数值计算运用Ｍａｔｈｅｍａｔｉｃａ可以象用计算器一样进行算术运算 ,但是与计算器不同的是 ,它能给出精确的结果 .系统直接支持有理数、和无理数的精确表示 ,例如 :Ｉｎ[1] :878/12 34Ｏｕｔ[1] :=4 396 17如果我们只需要得到近似值 ,可以使用Ｎ函数 .它的基本格式是 :Ｎ{表达式 ,精度ｎ}其结果用十进小数记数法表示 ,例如 :Ｉｎ[2 ] :=Ｎ[% ,10 ]Ｏｕｔ[2 ] :=0 .71150 72 934…     

亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

文[4]对简单形式的微分多项式fkf’+a的零点分布进行了讨论,文[1]对一般形式的微分多项式fkQ[f]+P[f]的零点分布进行了讨论.但由于极点给证明带来的困难,这些工作主要是对整函数来做的.本文证明了任一满足δ(∞,f)>k+2ΓQ+3ΓP+2/2k+2ΓQ+1的超越亚纯函数f,微分多项式fkQ[f]+P[f]在不含f,Q[f]极点和P[f]零、极点的可数个圆盘并集之外有无穷多个零点,其中k≥3Γp+2,而ΓQ,ΓP分别是f的微分多项式Q[f],P[f]的权.文[1]和[2,4,6]中的结论是本文结论的特殊情况.     

凸函数和凹函数的幂平均不等式

文 [1 ]获得了当 α≥ 1时的凸函数的幂平均不等式 (3)、(4 ) [1] .本文指出文 [1 ]中的一个错误 ,并且得到了 α≤ 1时的凹函数的幂平均不等式 .修正和充实了文 [1 ]的定理 .同时讨论了当 α取其它值时不等式的情况 .     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

曲线拐点充分条件证明中的常见错误

文[1]给出了判别曲线拐点的两个充分条件,文[2]给出了一个充分条件,但三个定理的证明都是错误的.同时,文[1]的两个推论也是错误的.本文通过反例分析了其错因,并给出了文[1]中一个拐点充分条件的正确证明.     

平均值函数的性质及应用

设函数 f (x)在 (-∞ , ∞ )上连续 ,当 x≠ 0时 ,我们称 F(x) =1x∫x0 f (t) dt为 f (x)在 [0 ,x]上的平均值函数 ,本文将介绍平均值函数 F(x)的若干性质并举例说明其应用 .一、F(x)的性质性质 1　 f(x)是 [0 ,x](或 [x,0 ])上的有界函数 ,F(x)也是 [0 ,x]或 [x,0 ]上的有界函数 .性质 2　若 f (x)为奇 (偶 )函数 ,则 F(x)也为奇 (偶 )函数 .性质 3　若 f(x)是周期为 T(T>0 )的周期函数 ,则limx→ ∞1x∫x0f (t) dt=1T∫T0f (t) dt (1 )　　　性质 4　若 f(x)为单调递增 (减 )函数 ,则 F(x)也为单调递增 (减 )函数 .性质 5　若对任意…     

亚纯函数微分多项式f~kQ[f]+P[f]的零点分布

从例外集的角度研究了亚纯函数微分多项式fkQ[f]+P[f]的零点分布,证明了:对于满足δ(∞,f)1-α>0的超越亚纯函数f(z),微分多项式fkQ[f]+P[f]在不含极点的可数个圆盘并集之外有无穷多个零点,其中k>1+ΓP+γP+α1(-1+αΓQ+ΓP-γP),ΓQ是Q[f]的权,ΓP,γP是P[f]的权和次数.本文推广了Hayman,Anderson,Langley等人的结论.     

对一道高考模拟题的思考

在高三教学中,笔者遇到了这样一道高考模拟题已知函数f(x)=ax-x4,x∈[1/2,1].若A,B是函数f(x)图象上任意不同两点,且kAB的取值集合为[1/2,4].求实数a的值.其参考答案为∵f′(x)= a-4x3,∴f′(x)在[1/2,1]上为减函数,∴f′(x)的值域为[a-4,a-1/2],又kAB的取值集合为[1/2,4],a-4=1/2,∴｛a-4=1/2,a-1/2=4,解得a=9/2.本文特记录笔者对该题目及其参考答案的真实思考过程,以供参考.     

Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

向量值双正交小波的存在性及滤波器的构造

引进了向量值多分辨分析与向量值双正交小波的概念.讨论了向量值双正交小波的存在性.运用多分辨分析和矩阵理论,给出一类紧支撑向量值双正交小波滤波器的构造算法.最后,给出4-系数向量值双正交小波滤波器的的构造算例.     

三元正交小波的构造

高维小波分析是分析和处理多维数字信号的有力工具.张量积小波有其自身的缺点.本文给出构造紧支撑三元不可分正交尺度函数和正交小波函数的新算法.当尺度函数的符号中含有因子1 z1221 z2221 z322的幂指数越高时,尺度函数越光滑.     

紧支撑正交的二维小波

基于Householder矩阵扩充,构造了紧支撑正交的二维小波,所构造小波函数的支撑不超过尺度函数的支撑,并且给出了容易实施的显式构造算法．另外,还通过构造反例说明Riesz定理不适用于二元三角多项式．最后,构造了算例．     

具有特殊伸缩矩阵的三元不可分正交小波的构造

多元小波分析是分析和处理多维数字信号的有力工具.不可分多元小波被广泛地应用在模式识别、纹理分析和边缘检测等领域.给出了构造具有伸缩矩阵(101-1-110-10)的紧支撑三元不可分正交小波的算法,利用该算法得到的小波函数继承了来源于尺度函数和符号函数的对称性和消失矩性质,从而为这类小波在信号处理方面的应用提供了便利.最后给出了数值算例.     

Wavelets from the Loop Scheme

Anewwavelet-based geometric mesh compression algorithm was developed recently in the area of computer graphics by Khodakovsky, Schröder, and Sweldens in their interesting article [23]. The new wavelets used in [23] were designed from the Loop scheme by using ideas and methods of [26, 27], where orthogonal wavelets with exponential decay and pre-wavelets with compact support were constructed. The wavelets have the same smoothness order as that of the basis function of the Loop scheme around the regular vertices which has a continuous second derivative; the wavelets also have smaller supports than those wavelets obtained by constructions in [26, 27] or any other compactly supported biorthogonal wavelets derived from the Loop scheme (e.g., [11, 12]). Hence, the wavelets used in [23] have a good time frequency localization. This leads to a very efficient geometric mesh compression algorithm as proposed in [23]. As a result, the algorithm in [23] outperforms several available geometric mesh compression schemes used in the area of computer graphics. However, it remains open whether the shifts and dilations of the wavelets form a Riesz basis of L₂(?²). Riesz property plays an important role in any wavelet-based compression algorithm and is critical for the stability of any wavelet-based numerical algorithms. We confirm here that the shifts and dilations of the wavelets used in [23] for the regular mesh, as expected, do indeed form a Riesz basis of L₂(?²) by applying the more general theory established in this article.     

紧支撑二元正交小波滤波器的构造

高维小波是处理多维信号的有力工具，张量积小波有其自身的缺点．本文给出矩形域上二元正交小波滤波器的一种参数化构造算法，二元小波滤波器的这种构造方法使我们能更方便地研究非张量积的二元正交小波．最后给出算例．     

A class of compactly supported orthogonal symmetric complex wavelets with dilation factor 3

When approximation order is an odd positive integer a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelets, one is symmetric and the other is antisymmetric about origin, are constructed explicitly. Additionally, when approximation order is an even integer 2, we also give a method to construct compactly supported orthogonal symmetric complex wavelets. In the end, there are several examples that illustrate the corresponding results.     

Biorthogonal M-band filter construction using the lifting scheme

The lifting scheme has been proposed as a new idea for the construction of 2-band compactly supported wavelets with compactly-supported duals. The basic idea behind the lifting scheme is that it provides a simple relationship between all multiresolution analyses sharing the same scaling function. It is therefore possible to obtain custom-designed compactly supported wavelets with required regularity, vanishing moments, shape, etc. In this work, we generalize the lifting scheme for the construction of compactly-supported biorthogonal M-band filters. As in the previous case, we used the flexibility of the scheme to exploit the degree of freedom left after satisfying the perfect-reconstruction conditions in order to obtain finite filters with some interesting properties, such as vanishing moments, symmetry, shape, etc., or that satisfy certain optimality requests required for particular applications. Moreover, for these lifted biorthogonal M-band filters, we give an analysis-synthesis algorithm which is more efficient than the standard algorithm realized with filters with similar compression capabilities. This revised version was published online in June 2006 with corrections to the Cover Date.