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1.
ON SIMULTANEOUS APPROXIMATION BY LAGRANGE INTERPOLATING POLYNOMIALS   总被引:1,自引:0,他引:1  
This paper considers to replace △_m(x)=(1-x~2)~2(1/2)/n +1/n~2 in the following result for simultaneousLagrange interpolating approximation with (1-x~2)~2(1/2)/n: Let f∈C_(-1.1)~0 and r=[(q+2)/2],then|f~(k)(x)-P_~(k)(f,x)|=O(1)△_(n)~(a-k)(x)ω(f~(a),△(x))(‖L_n-‖+‖L_n‖),0≤k≤q,where P_n( f ,x)is the Lagrange interpolating polynomial of degree n+ 2r-1 of f on the nodes X_nU Y_n(see the definition of the text), and thus give a problem raised in [XiZh] a complete answer.  相似文献   

2.
3.
Let Wβ(x)=exp(-1/2|x|β)be the Freud weight and pn(x) ∈пn be the sequence of orthogonal polynomials with respect to W2β(x),that is,∫∞-∞pn(x)pm(x)W2β(x)dx={0,1, n≠m, n=m.It is known that all the zeros of pn(x)are distributed on the whole real line.The present paper investigates the convergence of Gr(u)nwald interpolatory operators based on the zeros of orthogonal polynomials for the Freud weights.We prove that,if we take the zeros of Freud polynomials as the interpolation nodes,then Gn(f,x)→,f(x),n→∞ holds for every x ∈(-∞,∞),where f(x) is any continous function on the real line satisfying |f(x)|=O(exp(1/2|x|β)).  相似文献   

4.
Given two positive constantsαandβ,we prove that the integral inequality∫_0~1 f~(α+β)(x)dx≥∫_0~1 f~α(x)x~βdx holds for all non-negative valued continuous functions f satisfying∫_x~1 f(t)dt≥∫_x~1 tdt for x∈[0,1]if and only ifα+β≥1.This solves an open problem proposed recently by Ngo,Thang,Dat,and Tuan.  相似文献   

5.
Chen  Lu  Lu  Guozhen  Zhu  Maochun 《中国科学 数学(英文版)》2021,64(7):1391-1410
The classical critical Trudinger-Moser inequality in R~2 under the constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for any τ 0,it holds that ■ and 4π is sharp.However,if we consider the less restrictive constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1,where V(x) is nonnegative and vanishes on an open set in R~2,it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality ■ when V is nonnegative and vanishes on an open set in R~2.As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u) in R~2,(0.1)where V(x)≥0 and vanishes on an open set of R~2 and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schr?dinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.  相似文献   

6.
Let(ξ_n)_(n=0)~∞ be a Markov chain with the state space X = {1, 2, ···, b},(g_n(x, y))_(n=1)~∞ be functions defined on X × X, and F_(m_n,b_n)(ω) =1 /b_n sum from k=m_n+1 to m_n+b_n g_k(ξ_(k-1), ξ_k).In this paper the limit properties of F_(m_n,b_n)(ω) and the generalized relative entropy density f_(m_n,b_n)(ω) =-(1/b_n) log p(ξ_(m_n,m_n+b_n)) are discussed, and some theorems on a.s. convergence for(ξ_n)_n=0~∞ and the generalized Shannon-McMillan(AEP) theorem on finite nonhomogeneous Markov chains are obtained.  相似文献   

7.
ON KANTOROVICH-STIELTJES OPERATORS   总被引:1,自引:0,他引:1  
Let ν be a finite Borel measure on[0,1]The Kantorovich-Stieltjes polynomials are de-fined byK_n ν=(n+1)N_(k,n)(nN),where N_(k,n)(x)=x~k(1-x)~(n-k)(x[0,1],k=1,2,…,n)are the basic Bernsteinpolynomials and I_(k,n):=[k/(n+1),(k+1)/(n+1)](k=0,1,…,n;nN).We prove that the maximaloperator of the sequence(K_n)is of weak type and the sequence of polynomials(K_n ν)con-verges a.e.on[0,1]to the Radon-Nikodym derivative of the absolutely continuous part of  相似文献   

8.
Let TA(f)=integral form n= to 1/2(P_~n(x) + P_b~n(x))dx and let TM(f)=integral form n= to P_((+b)/2)~(n+1)(x)dx, where P_c~n denotes theTaylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of theTrapezoidal rule and the midpoint rule, respectively. and are each exact for all polynomial of degree ≤n+1.We let L(f) = αTM(f) + (1-α)TA(f), where α =(2~(n+1)(n+1))/(2~(n+1)(n+1)+1), to obtain a numerical integrationrule L which is exact for all polynomials of degree≤n+3 (see Theorem l). The case n = 0 is just the classicolSimpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacingP_(+b)/2)~(n+1)(x) by the Hermite cabic interpolant at a and b. we obtain some known formulae by a different ap-proach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by takingpiecewise polynomials of odd degree, each piece being the Taylor polynomial off at a and b. respectively. Ofcourse all of our formulae can be compounded over subintervals of [a, b].  相似文献   

9.
Let f(x) be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r(n) denote the number of solutions x of the congruence f(x) ≡ 0(mod n) satisfying0 ≤ x n. Define ?(x) =Σ 1≤n≤x r(n)-αx, where α is the residue of the Dedekind zeta function ζ(s, K) at its simple pole s = 1. In this paper it is shown that ∫_1~X?~2(x)dx? ε{X~(3-6/m+3+ε)if m ≥ 3,X~(2+ε) if m = 2,for any non-Abelian polynomial f(x) and any ε 0. This result constitutes an improvement upon that of Lü for the error terms on average.  相似文献   

10.
Let a(n)be the Fourier coefficients of a holomorphic cusp form of weightκ=2n≥12 for the full modular group and A(x)=∑_(n≤x)a(n).In this paper,we establish an asymptotic formula of the fourth power moment of A(x)and prove that ∫T1A~4(x)dx=3/(64κπ~4)s_4;2()T~(2κ)+O(T~(2κ-δ_4+ε))with δ_4=1/8,which improves the previous result.  相似文献   

11.
本文讨论了关于以下内积的正交多项式 :〈p(x) ,r(x)〉( u0 ,u(α,β) ) =∑∞k=0p(qk) r(qk) (qk-c) ak(b) k(q) k +λ∑∞k=0(Dqp) (qk) (Dqr) (qk) (aq) k(bq) k(q) k给出了它的一些代数性质以及和小 q-Jacobi多项式的关系 ,得到了在 C\([0 ,1 ]∪ H )的紧子集上Qn(x)P(α- 1,β- 1)n (x) n和 Pn(x)P(α- 1,β- 1)n (x) n的相对渐近性质 .其中 Qn(x)是 n次的小 q -Jacobi-Sobolev正交多项式 ,P(α- 1,β- 1)n (x)和 Pn(x)分别是关于线性泛函 u(α- 1,β- 1)和 u0 的首一的 n次正交多项式 .  相似文献   

12.
一类非线性微分方程空间周期解的存在性及唯一性   总被引:5,自引:1,他引:4  
利用Bendixson-Dulac定理,讨论系统  相似文献   

13.
本文给出了p—级数与广义积分∫10lnk-1x1-xdx,∫10lnk-1x1+xdx,∫10lnk-1x1-x2dx,∫10lnk-1x1+x2dx之间的关系.并通过一些p—级数的求和,给出了上述广义积分中某些积分的积分值.  相似文献   

14.
本文证明了勒让德多项式 Pn( x)的 k阶导数 P( k)n ( x)是 [-1 ,1 ]上关于权函数 ρ( x) =( 1 -x2 ) k的正交多项式 ,推广了 [1 ]的结果 .  相似文献   

15.
In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product
$ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1 $ \left\langle p,q\right\rangle _{S}=\int_{0}^{\infty }p(x)q(x)x^{\alpha }e^{-x}dx+Np^{\prime }(a)q^{\prime }(a),\alpha >-1  相似文献   

16.
张勇 《数学进展》2021,(2):184-194
设b,c为整数,定义广义中心三项式系数Tn(b,c)=[xnx2+bx+c]n=[π/2]∑k=0(n 2k)(2k n)bn-2kck(n∈N={0,1...}),这里[xn]P(x)表示多项式P(x)中xn项的系数.特别地,中心Delannoy多项式Dn(x)=Tn(2x+1,x2+x)(n∈N),中心三项式系数Tn=Tn(1,1)(n∈N).本文研究了孙智伟在[南京大学学报:数学半年刊,2019,36(1):1-99]中提出的猜想,即完全证明了两个关于Dn(x)和的超同余式和一个关于中心三项式系数的超同余式的特殊情形.例如,设p为素数,r,m为正整数满足p■m条件.则对于任何p-adic整数x,有1/m2p3r-3(prm-1∑k=0(2k+1)Dk(x)2-P2pr-1m-1∑k=0(2k+1)Dk(x)2)=0(mod p3).  相似文献   

17.
We characterize the class of ultraspherical polynomials in between all symmetric orthogonal polynomials on via the special form of the representation of the derivatives by

  相似文献   


18.
Following our earlier research, we propose a new method for obtaining the complete Pade table of the exponential function. It is based on an explicit construction of certain Pade approximants, not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and nontrivial coincidence is proved more generally for type II simultaneous Pade approximants for a family with distinct complex a's and we recover Hermite's classical formulas. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesu, Coussement, and van Assche, which generalize the classical Charlier orthogonal polynomials.  相似文献   

19.
利用函数zp(-1相似文献   

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