基于一类新的半离散的小波变换的信号重构公式

在基小波ψ(t)为带限的条件下,基于半离散的小波变换(Wψf)(na,a)(n∈Z,a∈R)信号f(t)的重构公式被给出.     

S—空间中正交小波基的一种构造法

本文提出了一种构造正交小波基{2~(m/2)ψ(2~mt-n),m,n∈Z}的方法.其中母函数ψ(t)∈S(急减函数空间),其Fourier变换ψ(w)∈C_0~∞.     

小波变换及其应用(续三)

两维正交小波基　在许多实际问题中 ,经常会遇到多维的信号 (如图像等 ) ,因此只有一维小波基是不够的。不难证明 (见 [1 ])若φ( t)与Ψ ( t)是小波标准正交基的尺度函数与母函数 ,φjm( x) =2 j2 φ( 2 jx -m) ,Ψjm( x) =2 j2 Ψ ( 2 jx -m) ,则 {Ψjn( x)φjm( y) ,φjn( x)Ψjm( y) ,Ψjn( x)Ψjm( y) }j,m,nz构成两维平方可积函数空间 (即满足∫∞-∞∫∞-∞|f ( x,y) |2 dxdy <∞的全体函数所构成的线性空间 )的一组标准正交基 (见图 1 2 ( a) ( b) )图 1 2(六 )离散小波变换的应用举例( 1 )消除信号中的噪音信号在生成和传输过程…     

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

SL(2,R)上一类极大算子的(H1∞,s,L1)型

本文我们引入了函数类Bδ(G//K)={ψ∈L1(G//K)‖ψ(t)|≤△-1(t)(1+t)1-δ,δ＞0},对f∈Lp(G//K),1≤p≤∞,和极大算子Mδf(x)=sup ε＞0 ψ∈Bδ(G//K) |ψε*f(x)|,证明了这类算子是(H1∞,s,L1)型的.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

Bounded Nonoscillatory Solutions of Nonlinear Functional Differential Equations

In this paper, we obtain some nonoscillatory theories of the functional differential equation （r（t）ψ（x（t））x （t）） ＋ f（t, x（t）, x（σ（t））） = 0, t ≥ t 0 , where r ∈ C 1 （[t 0 , ∞）; （0, ∞））, ψ∈ C 1 （R, R） and f ∈ C（[t 0 , ∞） × R × R, R）.     

分割区间来解题,也是一种好方法!

高考题1(2010.福建.理.15)已知定义域为(0,+∞)的函数f(x)满足:(1)对任意x∈(0,+∞),恒有f(2x)=2f(x)成立;(2)当x∈(1,2]时,f(x)=2-x.给出如下结论:①对任意m∈Z,有f(2m)=0;②函数f(x)的值域为[0,+∞);③存在n∈Z,使得f(2n+1)=9;④"函数f(x)在区间(a,b)上单调递减"的充     

ON THE METHOD OF SOLUTION FOR A KIND OFNONLINEAR SINGULAR INTEGRAL EQUATION

The solutions of the nonlinear singular integral equation ψo(t)2 2b/πi ∫L ψ(τ)/T-t dr =f(t), t ∈ L, are considered, where L is a closed contour in the complex plane, b ≠- 0 is a constant and f(t) is a polynomial. It is an extension of the results obtained in [1] when f(t) is a constant. Certain special cases are illustrated.     

小波变换及其应用(续一)

下面我们通过一个典型的例子来说明小波 Ψab( t)的频谱特征与参数 a的关系。例 1　设θ( t) =12πe- t22 ,令Ψ ( t) =-d2dt2 θ( t) =12π( 1 -t2 ) e- t22　　由于Ψ ( t)有因子 e- t22 ,所以当 |t|→∞时 ,Ψ ( t)趋于零的速度是非常快的 ,因此满足条件( A) ,又因∫∞-∞Ψ ( t) dt=-te- t22 |∞-∞ =0 ,故条件 ( B)亦满足。现取此Ψ ( t)为小波母函数。通常形象化地称此函数为“墨西哥帽子”(见图 4( a) )。不难求得 Ψ( t)的频谱 Ψ( ω) =ω2 e- -ω22 (见图 4( b) )。图 4现在来考察当参数 a,b变化时 ,Ψab( t)与Ψab(ω)变化…     

带参数的一阶泛函差分方程的定号周期解

In this paper,the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation ?u(n) = a(n)u(n)-λb(n)f(u(n-τ(n))),n ∈ Z by using global bifurcation techniques,where a,b:Z → [0,∞) are T-periodic functions with ∑T n=1 a(n) 0,∑T n=1 b(n) 0;τ:Z → Z is T-periodic function,λ 0 is a parameter;f ∈ C(R,R) and there exist two constants s_2 0 s_1 such that f(s_2) = f(0) = f(s_1) = 0,f(s) 0 for s ∈(0,s_1) ∪(s_1,∞),and f(s) 0 for s ∈(-∞,s_2) ∪(s_2,0).     

Maximal integral over observable measures

Let f be a continuous transformation on a compact,finite-dimensional manifold M,andψa continuous function on M.This paper establishes the following formula:ess sup lim sup n→∞ 1/n ψ_n(x)=sup{∫ψdμ︱μ∈O_f}≤lim sup n→∞ 1/n ess sup ψ_n(x),where ess sup denotes the essential supremum taken against the Lebesgue measure,ψ_n(x)=n-1 ∑ i=0 ψ(f~ix) and O_f is the set of observable measures.Examples are provided to illustrate that the inequality could be an equality or strict.Moreover,if μ is the unique maximizing observable measure for ψ,it is weakly statistical stable.     

Banach空间中渐近非扩张映射的不动点迭代

设D是一致凸Banach空间X的非空闭凸子集 ,T∶D→D是渐近非扩张映射且kn ≥ 1 ,∑ ∞n =1(kn- 1 ) <∞ .设T的不动点集F(T) ≠ ,T是全连续的 (X满足Opial条件 ) ,{xn},{yn},{zn}由定义 2给出 ,如果 ∑∞n =1cn <∞ ,∑ ∞n =1c′n <∞ ,∑ ∞n =1c″n <∞ ,且下列条件之一满足 :(i)b″n ∈ [a ,b] ( 0 ,1 ) ;b′n ∈ [0 ,β];bn ∈[0 ,α],αβ β <1 ;(ii)b′n ∈ [a ,b] ( 0 ,1 ) ;b″n ∈ [a ,1 ];bn ∈ [0 ,b];(iii)bn ∈[a ,b] ( 0 ,1 ) ;b′n ∈ [a ,1 ],则 {xn},{yn},{zn}强收敛于T的不动点 .( {xn}弱收敛于T的不动点 ) .     

Unb ounded Motions in Asymmetric Oscillators Dep ending on Derivatives

In this paper, we consider the unboundedness of solutions for the asymmetric equation x00+ax+?bx?+?(x)ψ(x0)+f(x)+g(x0)=p(t), where x+ = max{x, 0}, x? = max{?x, 0}, a and b are two different positive constants, f (x) is locally Lipschitz continuous and bounded,?(x), ψ(x), g(x) and p(t) are continuous functions, p(t) is a 2π-periodic function. We discuss the existence of unbounded solutions under two classes of conditions: the resonance case √1a+ √1b ∈Q and the nonresonance case√1a + √1b /∈Q.     

Z／（2∧e）上本原序列不同压缩映射的导出序列

设f(x）是Z/（2∧e）上n次强本原多项式,对形如xe-1 η（x0,…,xe-2）的二个e元布尔函数φ（xo,…,xe-1）和ψ（x0,…,xe-1）及二条序列a,b∈G（f（x)）e,若φ（a0,…,ae-1）=ψ（b0,…,be-1）,给出了函数φ（x0,…,xe-1）和ψ（x0,…,xe-1）之间的关系与序列a和b之间的关系,所给出的结论进一步说明了导出的二元序列具有良好的密码性质。     

A trichotomy for a class of equivalence relations

Let Xn, n ∈ N be a sequence of non-empty sets, ψn : Xn2 → IR+. We consider the relation E = E((Xn, ψn)n∈N) on ∏n∈N Xn by (x, y) ∈ E((Xn, ψn)n∈N) <=>Σn∈Nψn(x(n), y(n)) < +∞. If E is an equiv- alence relation and all ψn, n ∈ N, are Borel, we show a trichotomy that either IRN/e1≤B E, E1≤B E, or E≤B E0. We also prove that, for a rather general case, E((Xn, ψn)n∈N) is an equivalence relation iff it is an ep-like equivalence relation.     

具有无限时滞的二阶微分方程解的存在性

§ 1.　IntroductionRecently ,thedifferentialequationswithdeviatingargumentswereusuallydiscussed(see[1 ],[4],[5 ]) .In [1 ],AGARWALRPandO’REGANDconsideredequationy″(t) =f(t,y(t) ,y(σ(t) ) ) ,　a.e .t∈ [0 ,1 ]y(t) =ψ(t) ,　　　　　　　　t∈ [-r ,0 ]y( 1 ) =a ,( )andtheydiscussedtheexistenceofatleastonesolutionforequation ( ) .Inthispaper ,weconsideramoregeneralequation-x″(t) =f(t ,xt) ,　t∈ [0 ,1 ]x(t) =ψ(t) ,　　　　t∈ ( -∞ ,0 ]x( 0 ) =x( 1 ) =0 ,( 1 .1 )andsomeexistencetheor…     

一类积分微分方程周期解的存在性和唯一性

本文考虑具连续时滞和离散时滞的非线性积分微分方程x'(t)=A(t,x(t))x(t)+∫-∞tC(t,s)x(s)ds+∑i=1i gi(t,x(t—τi(t)))+b(t)和x’(t)=f(t,x(t))+∫-∞tC(t,s)x(s)ds+∑i=1igi(t,x(t-τi(t)))+b(t)周期解的存在性和唯一性问题,这里t∈R,x∈Rn;A(t,x),C(t,s)为n×n阶连续的函数矩阵; f(t,x),gi(t,x)(i=1,2,…,l),b(t)是n维连续向量.通过利用线性系统指数型二分性理论和泛函分析方法研究上述系统,获得了保证其周期解存在性、唯一性的充分性条件.我们除了实质性的推广和改进了已有的结果外,还得到三个新的定理,这是用已有的方法无法获得的(见文[1-30]).     

由Lagrange展开产生的一对公式

Let Γ denote the ring of formal power series over the complex field C. Let φ∈Γ with φ(0)=0 and φ′(0)=0 So that it has a compositional inverse ψ with ψ(φ(t))=φ(ψ(t))=1. Then for every f∈Γ and α∈C With α≠0 we have Lagrange′s formal expansion     

不规则小波框架的膨胀列

研究了L2(R)中小波框架{ψj,k}j,k={sjψ(sj·-kb)}j,k∈Z的膨胀列{sj}j的性质.如果{ψj,k}j,k是L2(R)的一个小波框架,那么膨胀列是无界的,在某些条件下{sj}j∈Z一定能够被重排为指标集Z上的一个非减数列,而且存在常数λ,μ∈(0,1)和p∈Z ,使得对j∈Z有λ相似文献