共查询到20条相似文献,搜索用时 109 毫秒
1.
Hu Ke 《数学年刊B辑(英文版)》1980,1(34):421-427
Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l}
{\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m},
\end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m.
Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l}
\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3,
\end{array}\], where \[\begin{array}{l}
{F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}).
\end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun.
Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l}
P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2},
\end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing. 相似文献
2.
M. M. Kabardov 《Vestnik St. Petersburg University: Mathematics》2009,42(3):169-174
The Euler-Knopp transformation is considered in terms of the problems of regularity and acceleration of the rate of convergence.
The object of study is the hypergeometric series
$
_n F_{n - 1} (a;b;z) = \sum\limits_{k = 0}^\infty {\frac{{(a_1 )_1 \cdots (a_n )_k }}
{{(b_1 )_k \cdots (b_{n - 1} )_k }}} \frac{{z^k }}
{{k!}} = \sum\limits_{k = 0}^\infty {\lambda _k z^k } .
$
_n F_{n - 1} (a;b;z) = \sum\limits_{k = 0}^\infty {\frac{{(a_1 )_1 \cdots (a_n )_k }}
{{(b_1 )_k \cdots (b_{n - 1} )_k }}} \frac{{z^k }}
{{k!}} = \sum\limits_{k = 0}^\infty {\lambda _k z^k } .
相似文献
3.
Using the following notation: C is the space of continuous bounded functions f equipped with the norm
, V is the set of functions f such that
, the set E consists of fCV and possesses the following property:
4.
The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞ 相似文献
5.
Н. В. Гуличев 《Analysis Mathematica》1984,10(1):3-13
Let L∞ denote the space of measurable 1-periodic essentially bounded functionsf(x) with ∥f∥=vrai sup ¦f(x)¦,S k (f, x) thek-th partial sum of the Walsh-Fourier series off(x),L k thek-th Lebesgue constant. The following theorem is proved. Theorem. Letλ={λ K } be a sequence of nonnegative numbers, $$\left\| \lambda \right\|_1 = \mathop \sum \limits_{k = 1}^\infty \lambda _k< \infty ,\left\| \lambda \right\|_2 = (\mathop \sum \limits_{k = 1}^\infty \lambda _k^2 )^{1/2} ,m = log[(\left\| \lambda \right\|_1 /\left\| \lambda \right\|_2 )]$$ .Then for an arbitrary function f∈L ∞ the following inequalities hold true $$\begin{gathered} \left\| {\mathop \sum \limits_{k = 1}^\infty \lambda _k \left| {S_k (f,x)} \right|} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - 2m]} + c)\left\| f \right\|, \hfill \\ \hfill \\ \mathop \sum \limits_{k = 1}^\infty \lambda _k \left\| {S_k (f)} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - m]} + c)\left\| f \right\| \hfill \\ \end{gathered} $$ , where[y] denotes integral part of a number y>0 and c is an absolute constant. A corollary of the above theorem is that for each functionfεL ∞ the Lebesgue estimate can be refined for a certain sequence of indices, while the growth order of Lebesgue constants along that sequence can be arbitrarily close to the logarithmic one. “In the mean”, however, the Lebesgue estimate is exact. A further corollary deals with strong summability. 相似文献
6.
LIN ZHENGYAN 《数学年刊B辑(英文版)》1981,2(2):181-185
In this paper, we consider a central limit theorem for the sequence of stationary m-dependent random variables, the variance of which is possibly infinite.
Theorem. Let {Xn, n=l, 2,...} be a sequence of stationary m-dependent random variables with means zero. The following conditions are satisfied.
(i) \[{M^2}\int_{{\text{|}}{X_1}| > M} {dP} /\int_{{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\]
(ii) \[\int_{\{ {X_1}| < M,|{X_i}| < M} {X_1^{}} {X_i}dP/\int_{|{X_1}| < M} {X_1^2} dP \to 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (M \to \infty )\]
then there are constants Bsub
7.
QUALITATIVE METHOD IN ANALYSIS OF A P-L-L
WITH TANGENT CHARACTERISTIC AND
FREQUENCY MODULATION INPUT
In analysis of p-L-L with tangent characteristic and frequency modulation input, we have obtained the following two types of the phase looked loop equation.
\[\begin{array}{l}
\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + \alpha \frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t + {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (I)\\frac{{{\partial ^2}\varphi }}{{\partial {t^2}}} + (\alpha + \eta {\sec ^2}\varphi )\frac{{d\varphi }}{{dt}} + \gamma \tan \varphi = {\beta _1} + {\beta _2}(\cos {\Omega _M}t - {\Omega _M}\sin {\Omega _M}t){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (II) \(\alpha > 0,\gamma > 0,\eta > 0,{\beta _1} > 0,{\beta _2} > 0,{\Omega _M} > 0)
\end{array}\]
In this paper, our aim is to explain the usual qualitative method and Lyapunov's function method, by which the existence of a periodic solution of (I), (II) is established. In addition, we especially point out: How is to construct the Lyapunovas function
for the nonlinear and nonairtoiiomous system? This is a very important problem. 相似文献
8.
Cao Jiading 《数学年刊B辑(英文版)》1981,2(2):243-255
In this article we generahze the polynomials of Kantorovitch \({P_n}(f)\) . Let \({B_n}\) be a sequence of linear operators from C[a,b] into \({H_n}\), if \[f(t) \in L[a,b],F(u) = \int_a^u {f(t)dt} ,{A_n}(f(t),x) = \frac{d}{{dx}}{B_{n + 1}}(F(u),x)\], here \({B_n}\)satisfy\[\begin{array}{l}
(a):{B_n}(1,x) \equiv 1,{B_n}(u,x) \equiv x;\(b):for{\kern 1pt} {\kern 1pt} g(u) \in C[a,b]{\kern 1pt} {\kern 1pt} we{\kern 1pt} {\kern 1pt} have{\kern 1pt} {\kern 1pt} {B_n}(g(u),b) = g(b).
\end{array}\]. we call such \({A_n}(f)\) generalized polynomials of Kantorovitch (denoted by \({A_n}(f) \in K\) ). Let
\[\begin{array}{l}
{\varepsilon _n}({W^2};x)\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}} \left| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right|,\{\varepsilon _n}{({W^2}{L^p})_{{L^p}}}\mathop = \limits^{def} \mathop {\sup }\limits_{f \in {W^2}{L^p}} {\left\| {{A_n}(f(t),x) - f(x) - f'(x)({A_n}(t,x) - x)} \right\|_p}.
\end{array}\]
We have proved the following results:
Let An he a sequence of linear continuous operators of type \[C[a,b] \Rightarrow C[a,b],{D_n}(x,z)\mathop = \limits^{def} {A_n}(\left| {t - z} \right|,x) - \left| {x - z} \right| - ({A_n}(t,x) - x)Sgn(x - z),{A_n}(1,x) = 1\] then (1):\({\varepsilon _n}({W^2};x) = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dz\), (2): Moreover, if \({A_n}\) be a sequence of linear positive operators, then for \(\left[ {\begin{array}{*{20}{c}}
{a \le x \le b}\{a \le z \le b}
\end{array}} \right]\) ,we have \({D_n}(x,z) \ge 0\), and \({\varepsilon _n}({W^2};x) = \frac{1}{2}{A_n}({(t - x)^2},x)\).
Let \({A_n}(f) \in K\) be a sequence of linear positive operators,\[{R_n}{(z)_L} = \frac{1}{2}\int_a^b {\left| {{D_n}(x,z)} \right|} dx\],then \[{R_n}{(z)_L} = \frac{1}{2}\left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right]\] and \[{\varepsilon _n}{({W^2}L)_L}{\rm{ = }}\frac{1}{2}\left\| {{B_{n + 1}}({u^2},z) - {z^2}} \right\|\]. Let \[{g_n} = \frac{1}{2}\mathop {\max }\limits_{a \le x \le b} {A_n}({(t - x)^2},x),{h_n} = \frac{1}{2}\mathop {\max }\limits_{a \le z \le b} \left[ {{B_{n + 1}}({u^2},z) - {z^2}} \right],\] then \[{\varepsilon _n}{({W^2}{L^p})_{{L^p}}} \le {g_n}^{1 - \frac{1}{p}}{h_n}^{\frac{1}{p}}(1 < p < \infty ).\] 相似文献
9.
We obtain the weighted sum identities for ■(-1)~kkζ(k,s-k),■k kζ(2k,2s-2k),■kkζ(2k+1,2s-2k-1),■k~2kζ(2k,2s-2k) and ■k~2kζ(2k+1,2s-2k-1). 相似文献
10.
Hansjörg Linden 《Integral Equations and Operator Theory》1992,15(4):568-588
In this paper operator functions of type
|