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1.
关于线性算子的高阶饱和   总被引:1,自引:1,他引:0  
设f(x)∈C_(2π)。而f(x)~sum from k=0 ( )A_k(f_1k)≡α_0/2 sum from k=1 ( )(α_kcoskx b_ksinkx)。 又设 U_n(f,x)=1/πintegral from -πto π(f(x t)u_n(t)dt,) 其中u_n(t)=1/2 sum from k=1ρ_k~(n)coskt满足条件: integral from 0 to k(|u_n(t)|dt=O(1),)ρ_k~(n)→1(n→∞;k=1,2,…,)。设m是正整数,ρ_0~(n)=1。记~mρ_k~(n)=sum form v=0 to ∞ ((-1)~(m~(-v))(m v)ρ_k v~(n) (k=0,1,…,)。)T.Nishishiraho考虑了在ρ_k~(n)=O(k>n)的情况下U_n(f,x)的饱和问题,证明了。 定理A 设{_n}是收敛于0的正数列,使得  相似文献   

2.
一、引言 设函数f∈c_(2x)的Fourier级数为 f(x)~(1/2)a_0+sum from k=1 to ∞(a_kcoskx+b_ksinkx),S_k(f,x)为其k阶部分和.又设ω(t)是一个连续模函数,且记 H~ω:={f|,ω(f,t)≤ω(t)},其中ω(f,t)是f的连续模.当ω(t)=Mt~α,(0<α≤1)时,则记H~ω=Lip_Mα.熟知对于任何f∈Lip_M~α,0<α<1,有M′使其共轭函数∈Lip_M′~α.  相似文献   

3.
1.符号与基本结果对对[0,1]上的可积函数f(x),Kantorovitch算子定义为: K_n(f,x)=(n+1)sum from k=0 to n(p_(n-K)(x)integral from ?(f(t)dt)其中p_(n-K)(x)=(n K)x~K(1-x)~(n-K),I_K=[K/(n+1),(K+1)/(n+1)]。记M(u)是N-函数,N(v)是其young意义下的余函数,用M(u)∈△_2表示,存在正数c,u_0满足  相似文献   

4.
Some new series inversion formulas of the general form F(n)=sum form k=0 to r(A_(k,m)f(n-mk)) if and only if f(n)=sum form k=0 to r(B_(k,n)F(r_o-mk)) valid for either r=[n/m]or r=∞ are presented. These relations generalize many of those given by the author in a long series of preceding papers. An interesting example is given by A_(k,n)=(-y)~kA_k(p-λn,t(1+λm)) and B_(k,n)=y~kA_k(p-λn,(1-t)(1+λm)) where A_k(a,b)=a/(a+bk) in terms of binomial coefficients. Here p,t,y and λ are arbitrary complex numbers. A corresponding Abel coefficient case occurs which uses numbers of the form a(a+bi)(i-1)/i!. An application to special functions studied by Singhal and Kumari is given, and it is also shown that sum form k=0 to ∞(z~kA_k(a+ck,b))=x~a(x-b(x-1))/(x-(b+c)(x-1)), where z=(x-1)x~(-b-c), with a corresponding case for the Abel coefficients sum from k=0 to ∞(z~kB_k(a+ck,b))=x~o(1-b logx)/(1-(b+c)log x),where z=(log x)x~(-b-c) From these expansions we then have easily the new convolution formula for Rothe coeffici  相似文献   

5.
■_n(f,x)=sum from k=0 to x(∫_(I_k)f(t)dt■_(nk)(x)),称为Meyer—Knig—Zell算子,其中 记  相似文献   

6.
设f(x)∈L_(2π)的Fourier级数为 f(x)~a_0/2+sum from n=1 to ∞ (a_ncosnx+b_nsinnx)sum from n=0 to ∞(A_n(f,x)) (1)以s_n(f,x)sum from i=0 to n(f,x)表示(1)第n部分和。称序列  相似文献   

7.
在这篇文章,我们对拟周期系统dx/dt=A(ω_1t,ω_2t.…,ω_mt)x (0.1)建立了Floquet理论.其中n×n方阵A(u_1,u_2,…,u_m)是u_1,u_2,…,u_m以2π为周期的周期方阵,同时假定A(u_1,u_2,…,u_m)∈C~(τ),τ=(N 1)τ_0,τ_0=2(m 1),N=1/2n(n 1).我们定义了(0.1)的特征指数根β_1,β_2,…,β_n,假设下式成立:(一)|sum from u=1 to m k_μω_μ|≥K(ω)(sum from μ=1 to m|k_μ|)~[-(1 m)](二)|sum form μ=1 to m ik_μω_μ sum form v=1 to n i_vβ_v|≥k(ω,β)(sum form μ=1 to m|k_μ|~[-(m 1)]其中K(ω),K(ω,β)>0,k_μ,i_v是整数,k_1,k_2…,k_m不全为零:i~2=-1|sum form μ=1 to n j_μ|≤1.sum form μ=1 ton|j_μ|≤2.那末有拟周期线性变换,把(0.1)化为常系数的线性系统.  相似文献   

8.
设a_0,a_1,…,a_n是实轴或复平面上任意n 1个点。记 ω_(j 1)(x)=multiply from v=0 to j(x-a_v)(j=0,1,…,n),ω_0(x)=1。 (1)以H_n(x)表示以a_0,…,a_n为节点的n次插值多项式, R_n(x)=f(x)-H_n(x)。 (2)对任意k=0,1,…,n关于R_n~((k))(x)用f限定阶数的差商(或导数)来表示的问题,我们在[1]中证明了等式  相似文献   

9.
卢旭光 《计算数学》1988,10(4):398-407
1.引言 用△_k是表示R~k中的单纯形:△_k={X=(x_1,x_2,…,x_k)∈R~k|x_i≥0,i=1,2,…,k;sum from i=1 to k(x_i)≤1};C(△_k)表示定义在△_k上的连续函数的全体.记||f||=||f||_(△_k):=sup|f(X)|,ω(f,t):=sup |f(X)-f(Y)|。连续函数ω(t),t∈[0,+∞)称为  相似文献   

10.
Let f(x)∈L_(2π) and its Fourier series by f(x)~α_0/2+sum from n=1 to ∞(α_ncosnx+b_nsinx)≡sum from n=0 to ∞(A_n(x)). Denote by S_n (f,x) its partial sums and by E_n~q(f,x) its Euler (E, q)-means, i. e. E_n~q(f,x)=1/(1+q)~π sum from m=0 to n((?)q~(n-m)S_m(f,x)), with q≥0 (E_n~0≡S_n). In [1] Holland and Sahney proved the following theorem. THEOREM A Ifω(f,t) is the modulus of continuity of f∈C_(2π), then the degree of approximation of f by the (E,q)-means of f is givens by##特殊公式未编改  相似文献   

11.
§0.引言为了下面解释的方便起见,我们首先给出如下几个定义:定义1 称一个连续实变复值函数φ(t)为一个非负定函数,如果对任何 n≥1,实数t_1,…,t_n 及复数λ_1,…,λ_n,有 sum from i,k=1 to n λ_iλ_kφ(t_i-t_k)≥0.而当φ(0)=1时,此φ(t)被称为标准非负定函数(实际上就是概率论中的特征函数).定义2 称非负定函数φ(t)是正则的,如果存在 f(x)∈L~1(-∞,∞),使φ(t)为f(x)的 L~1-Fourier 变换.而称产 f(x)为φ(t)的密度函数.定义3 设 g(t)是 L~1(-∞,+∞)中某函数的 L~1-Fourier 变换,若  相似文献   

12.
Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e~(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ~(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i~2=1ω_i~(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~p(ν_ω) if p0 p1, p2 ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.  相似文献   

13.
何纯瑾 《数学杂志》1990,10(1):59-60
本文求解形为,f(x)=multiply from k=1 to 2(x~2-p_kx-q_k)+k multiply from k=1 to 2(x~2-r(?)x-s_k) (1)(其中 n 为偶数)或 f(x)=multiply from k=1 to n(x-Pk)+K multiply from k=1 to n(x-q_k) (2)的“乘积多项式”的所有二次因式 x~2-u(?)x-v_i.用[1]中方法,得初始近似因子ω(x)=x~2+ux+v.再分两步求ω(x)的修正因子ω(x):1.用ω~2(x)除 f(x),得余式 R_1(x);2.用ω~2(x)除 xR_1(x),得余式 R_2(x).再取 R_1(x)与 R_2(x)的适当线性组合,消去  相似文献   

14.
1 引言设函数f(z)在单位园|z|≤1内解析。记n(ω)=n(ω),D,f)为f(z)=ω在D内解的个数。若P(R)=1/2π integral from n=0 to 2x(n(Re~(iθ))dθ≤P),则称此函数为D内的平均P叶函数。特别,当P=1时,  相似文献   

15.
设d≥1为正整数,S为Rd中的单纯形,C(S)为S上连续函数类,f(x)∈C(S),f(x)≥0,f(x) 0,p>1,‖@‖p为通常的Lp范数,‖@‖为一致范数,则存在Pn(x)∈∏+n,d={Pn(x)Pn(x)=ak≥0},常数C>0使‖f-1/Pn‖p≤C[ω2φ(f,/4n)+‖f‖/n],这里对k,x∈Rd,k=(k1,k2,…,kd),x=(x1,x2,…,xd),记|k|=k1+k2+…+kd,|x|=x1+x2+…+xd,xk=xk11xk22…xk11dk22,ω24(f,t)为单纯形S上关于一致范数的二阶Ditzian-Totik光滑模.  相似文献   

16.
Let a(x)=(a_(ij)(x)) be a uniformly continuous, symmetric and matrix-valued function satisfying uniformly elliptic condition, p(t, x, y) be the transition density function of the diffusion process associated with the Diriehlet space (, H_0~1 (R~d)), where(u, v)=1/2 integral from n=R~d sum from i=j to d(u(x)/x_i v(x)/x_ja_(ij)(x)dx).Then by using the sharpened Arouson's estimates established by D. W. Stroock, it is shown that2t ln p(t, x, y)=-d~2(x, y).Moreover, it is proved that P_y~6 has large deviation property with rate functionI(ω)=1/2 integral from n=0 to 1<(t), α~(-1)(ω(t)),(t)>dtas s→0 and y→x, where P_y~6 denotes the diffusion measure family associated with the Dirichlet form (ε, H_0~1(R~d)).  相似文献   

17.
设f(x)∈C_(2π)。本文讨论两种线性算子对f(x)的逼近,全文分两个部分。 在第一部分中,我们考虑在正卷积型三角多项式线性算子中占重要地位的Fejr-Korovkin算子K_n(f,x)=1/π integral from -x to π (f(x+t)k_n(t)dt),其中k_n(t)≡1/2+sum from k=1 to n (ρ_k~((n)) cos kt)=1/2+sum from k=1 to n (F_n(k/n+2)coskt),F_n(x)=(1-x)cosπX+1/n+2 cot π/n+2·sinπx.由于它满足Korovkin条件:所以有下述结果:设f(x)∈C_(2π),f″(x)∈C_(2π)。那么,当n→∞时,成立着  相似文献   

18.
本文考虑随机幂级数:f(z,ω)=sum from n=0 to ∞ a_n e~(iω_n)z~n (1.1)其中 a_n≥0(n=0,1,…),{ω_n}是概率空间(Ω,(?),P)上的 steinhaus 序列。我们给出了f(z,ω)a.s.属于 α-Bloch 函数类(?)~α,(?)_0~α的条件,当α=1时,得出[1]中相应的结果。  相似文献   

19.
设p_m≥0↓,sum from k=0 to n(p_n)=P_m,n=0,l,…,p_0=P_0=1,P_n→∞(n→∞)若N_n=1/P_n sum from k=0 ton(p_(n,k)S_k→S(n t。0→∞)),则说{S_k}关于算子(N,p_n)收敛于S.设f(x)∈L_(?),S_n(f,x)为  相似文献   

20.
施咸亮 《数学学报》1980,23(2):192-202
<正> 设 f(x)是周期 2π的 L 可积函数,(?)[f]=a_0/2+sum from v=1 to ∞ a_v cosvx+b_vsin vx为其富里埃级数.本文的目的是对于 L_p(1≤p≤∞)中的函数估计量  相似文献   

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