共查询到20条相似文献,搜索用时 562 毫秒
1.
HU KE 《数学年刊A辑(中文版)》1981,2(1):21-24
Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\]
\[\begin{gathered}
\frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\
\end{gathered} \]
Milin-Lebedey proved that
\[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \]
where p>l and \[\lambda \]>0.
In this paper, we have proved the following theorems;
Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and
\[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\]
then F(x) is a decreasing function of x on [0, 1].
This theorem is stronger than the result (1).
Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and
\[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \]
then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2,...)In the case p=2 this is contained in the Miiin-Lebedev's result. 相似文献
2.
HU KE 《数学年刊B辑(英文版)》1981,2(1):21-24
Let \[\varphi (x) = \sum\limits_{k = 1}^\infty {{A_k}} {x^k},\Phi (x) = {e^{\varphi (x)}} = \sum\limits_{k = 1}^\infty {{D_k}} {x^k}\]
\[\begin{gathered}
\frac{1}{{{{(1 - x)}^\lambda }}} = \sum\limits_{k = 1}^\infty {{d_k}} (\lambda ){x^k} \hfill \ {\overline \Delta _n}(\lambda ) = {\lambda ^{2 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} \mathop {|{A_k}|}\nolimits_{}^p - \sum\limits_{k = 1}^\infty {\frac{1}{k}} \hfill \\
\end{gathered} \]
Milin-Lebedey proved that
\[\sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^{p - 1}(\lambda )}}} \leqslant \exp \{ {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}} |{A_k}{|^p}\} \]
where p>l and \[\lambda \]>0.
In this paper, we have proved the following theorems;
Theorem 1. Let \[p \geqslant 1,\lambda > 0\] and
\[F(x) = \sum\limits_{k = 0}^\infty {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}} {x^p}\exp \{ - {\lambda ^{1 - p}}\sum\limits_{k = 1}^\infty {{k^{p - 1}}|{A_k}{|^p}{x^k}} \} (2)\]
then F(x) is a decreasing function of x on [0, 1].
This theorem is stronger than the result (1).
Theorem 2. Let \[p \geqslant 2,\lambda > 0\] and
\[{{\bar Q}_n}(\lambda ) = \frac{1}{{n + 1}}\sum\limits_{k = 0}^n {\frac{{|{D_k}{|^p}}}{{d_k^p(\lambda )}}\exp } \{ - \frac{1}{{n + 1}}\sum\limits_{v = 1}^n {\overline {{\Delta _p}} } (\lambda )\} \]
then \[{{\bar Q}_n}(\lambda )\] is a decreasing fimctLon of n(n=l, 2,...)In the case p=2 this is contained in the Miiin-Lebedev's result. 相似文献
3.
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5.
在任意实的Banach空间中研究了用具误差的修正的Ishikawa与Mann迭代程序来逼近一致L-Lipschitz的渐近伪压缩映象不动点的强收敛性问题,在去掉条件$$\sum\limits_{n=0}^{\infty}\alpha_{n}^{2}<\infty, \q \sum\limits_{n=0}^{\infty }\gamma_{n}<\infty,\q \sum\limits_{n=0}^{\infty }\alpha_{n}(\beta_{n}+\delta_{n})<\infty,\q \sum\limits_{n=0}^{\infty}\alpha_{n}(k_{n}-1)<\infty$$之下,证明了相关文献的结果仍然成立.所得结果不但改进和推广了最近一些人的最新结果,而且也从根本上改进了定理的证明方法. 相似文献
6.
SUN YONGSHENG 《数学年刊B辑(英文版)》1980,1(2):273-282
Given a sequence of positive real numbers \[{\varepsilon _0},{\varepsilon _1},...,{\varepsilon _n},...\] which satisfy the
conditions \[{\varepsilon _v} \to 0,{\varepsilon _v} - {\varepsilon _{v + 1}} \ge 0,{\varepsilon _v} - 2{\varepsilon _{v + 1}} + {\varepsilon _{v + 2}} \ge 0\] for v =0, 1, 2, ..., and a class L(s)
of 2pi-periodic, L-integrable functions f(x) such that \[{E_n}{(f)_L} \le {\varepsilon _n}(n = 0,1,2,...)\],
where \[{E_n}{(f)_L}\] is the best mean approximation of f(x) by trigonometrical polynomials
of degree ≤n Let \[{S_n}(f)\] be the n-th partial sum of the Fourier series of f(x). It’s
known that Oskolkov has proved \[\mathop {\sup }\limits_{f \in L(\varepsilon )} ||f - {S_n}{(f)_L}|| = \sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} \] where \[||f|{|_L} = \int_0^{2\pi } {|f(x)|} dx\] Oskolkov asked whether there is a single function \[{f_0}(x) \in L(s)\] for which the above relation is satisfied for all n, In this paper the following result is obtained.
Theorem Let \[L(\varepsilon )\] be a class of 2pi-periodic, L-integrable functions as giyen above, then there exists a funotion \[{f_0}(x) \in L(\varepsilon )\] such that \[{{\tilde f}_0}(x) \in L(\varepsilon )\] and
\[\begin{array}{l}
\overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{{{\left\| {{f_0} - {S_n}({f_0})} \right\|}_L}}}{{\sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} }} \ge C > 0\\overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{{{\left\| {{{\tilde f}_0} - {S_n}({{\tilde f}_0})} \right\|}_L}}}{{\sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} }} \ge C > 0
\end{array}\]
where C is an absolute constant. Some generalizations of the theorem are given. 相似文献
7.
WANG JIAGANG 《数学年刊A辑(中文版)》1981,2(1):13-20
若说\[(\Omega ,\mathcal{F},P)\]为完备概率空间,\[F = {({\mathcal{F}_t})_{t \in [a,b]}}\]为\[\mathcal{F}\]的递增子\[\sigma \]域族,且满足通常
条件,\[b \leqslant \infty \].又\[W = \{ {W_t},0 \leqslant t \leqslant b\} \]为关于F的Wiener过程,\[X = \{ {X_t},0 \leqslant t < b\} \]为
循序讨测过程,且
\[P\{ \int_0^b {X_t^2} dt < \infty \} = 1\],
则可定义X关于W的Ito随机积分
\[{(X \cdot W)_t} = \int_0^t {{X_s}} d{W_s},0 \leqslant t \leqslant b\]
这时若记
\[{Z_t} = \exp \{ \int_0^t {{X_s}} d{W_s} - \frac{1}{2}\int_0^t {{X_s}^2} ds\} \]
它便是一个指数(局部)鞅.本文的目的在于证明当X为循序可测正态过程时,只要X关于W的积分存在,\[{\text{\{ }}{Z_t}0 \leqslant {\text{t < b\} }}\]总是一致可积的。
引理1若\[\{ {Z_t},0 \leqslant t < b\} \]为实可测正态过程且
\[\int_0^{\text{b}} {\left\| {{X_t}} \right\|} d{m_t} < \infty \]
其中\[\left\| {{X_t}} \right\| = {(E|{X_t}{|^2})^{1/2}}\],\[{m_t}\]为[0,b)上右连续递增函数,则X的几乎所有样本函数关于\[{m_t}\]可积,且其轨道积分
\[\tilde I = \int_0^{\text{b}} {{X_t}} d{m_t}\]
为正态分布随机变量.
引理2若\[X = \{ {X_t},0 \leqslant t < b\} \]为可测正态过程,其几乎所有样本函数关于右连续增函数\[{m_t}\]可积,即
\[P(\int_0^b {|{X_t}} |d{m_t} < \infty ) = 1\]
则按轨道积分 \[\tilde I = \int_0^{\text{b}} {{X_t}} d{m_t}\]
是正态分布随机变量.
引理3 若\[\{ {\xi _n},n \geqslant 1\} \]为正态分布随机变量序列,则
\[\sum\limits_{j = 1}^\infty {E{\xi _i}^2} \leqslant {[Eexp( - \frac{1}{2}\sum\limits_{j = 1}^\infty {{\xi _i}^2} )]^{ - 2}}\]
进而若\[\sum\limits_{j = 1}^\infty {E{\xi _i}^2} < 1\],则
\[E[exp(\frac{1}{2}\sum\limits_{j = 1}^\infty {{\xi _i}^2} )] \leqslant {(1 - \sum\limits_{j = 1}^\infty {E{\xi _i}^2} )^{ - \frac{1}{2}}}\]
引理4若\[{m_s}\]为[0, b)上右连续增函数,又\[X = \{ X_t^{(i)},0 \leqslant t < b,1 \leqslant i < \infty \} \]为正态
过程,则当\[P\{ \sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < \infty \} = 1\]时必有
\[\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < \infty \} = 1\]
进而若;\[\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_t}} } < 1\],必有
\[Eexp(\frac{1}{2}\sum\limits_{i = 1}^\infty {\int_0^b {{{({X_t}^{(i)})}^2}d{m_s}} } ) \leqslant {(1 - \sum\limits_{j = 1}^\infty {E\int_0^b {{{({X_t}^{(i)})}^2}d{m_s}} } )^{ - \frac{1}{2}}}\]
定理 若\[W = (W_t^{(1)},...,W_t^{(n)},...)\]为一个具有无限个分量的过程,其分量都是连续
正态独立增量过程且满足
\[\begin{gathered}
E\{ W_t^{(i)} - W_s^{(i)}\} = 0 \hfill \ E\{ (W_t^{(i)} - W_s^{(i)})(W_t^{(j)} - W_s^{(j)})\} = {\delta _{ij}}(m_t^{(i)} - m_s^{(i)}) \hfill \\
\end{gathered} \]
又\[\{ {f_t} = (f_t^{(1)},...,f_t^{(n)},...)\} \]为循序可测正态过程,若
\[P\{ \sum\limits_{i = 1}^\infty {\int_0^b {{{({f_t}^{(i)})}^2}dm_t^{(i)}} } < \infty \} = 1\]
则 \[{Z_t} = \exp \{ \sum\limits_{i = 1}^\infty {\int_0^b {{f_s}^{(i)}dW_s^{(i)} - \frac{1}{2}\int_0^t {{{({f_s}^{(i)})}^2}dm_s^{(i)}} } } \} ,0 \leqslant t < b\]
是一致可积鞅,特别有\[E{Z_0} = 1\]
利用上述结果及正态过程的Hida-Cramer分解,可以象[1]一样方便地讨论正态测
度的等价性问题并求出其Radon-Nikodym导数. 相似文献
8.
Wang Kunyang 《数学年刊B辑(英文版)》1982,3(6):789-802
Let Q_N={\bar x=(x_1,\cdots ,x_N)|-pi \leq x_i <\pi,i=1,\cdots,N} and X(Q_N) denote L(Q_N) and C(Q_N) , The square de la УаДбо Poussin sums of f\in X (Q_N) are defined by
$V_n^n+l(f;\bar x)=\frac{1}{\pi ^N}\int _Q_N f(\bar x+\bar t)\prod\limits_{i = 1}^N {(\frac{1}{{l + 1}}} \sum\limits_{v = n}^{n + l} {{D_v}({t_i}))d\bar t(n,l = 0,1,2, \cdots )}$
where D_v(t) =sin(v+1/2)t/2sint/2, - The differences $R_n,l(f;\bar x)=f(\bar x)-V_n^n+l(f;\bar x)$ are called square remainders. We denote by E_k(f)_X the best approximation of the function f\in X(Q_N) by N-multiple trigonometric polynomials of order K.
Theorem Let {\varepsilon _k}_k=0^\infty be a sequence such that \varepsilon _n \downarrow \infty(n\rightarrow \infty), the class $X(\varepsilon)={f\in X(Q_N)|E_k(f)_X \leq \varepsilon _k,k=0,1,2,\cdots}$ Then
$C_N^'\sum\limits_{v=0}^n+l \frac {\varepsilon_v+nln^N-1(3+v/(l+1))}{v+l+1}\leq sup_{f\in X(\varepsilon)||R_n,l(f)||_X\leq C_N \sum\limits_{v=0}^{n+l}\frac {\varepsilon _v+nln^N-1(3+v/l+1)}{v+l+1}$
where C_N>C'_N>0 are constants depending only on N. 相似文献
9.
设$E$为一致光滑Banach空间,$A:E\to E$为有界次连续广义${\it \Phi} $-增生算子满足:对任意$x_0\in E$,选取$m\ge 1$,使得$\| x_0 - x^* \| \le m$且$\mathop {\underline {\lim } }\limits_{r \to \infty } {\it \Phi} (r) > m\left\| {Ax_0 } \right\|$.设$\{C_n\}$为$[0,1]$中数列满足控制条件: i)$C_n\to 0\,(n\to\infty)$; ii)$\sum\limits_{n = 0}^\infty {C_n } = \infty $.设$\{x_n\}_{n\ge0}$由下式产生x_{n + 1} = x_n - C_n Ax_n ,\q n \ge 0, \eqno{(@)}$$则存在常数$a>0$,当$C_n < a$时,$\{x_n\}$强收敛于$A$的唯一零点$x^{*}$. 相似文献
10.
设f是区间[a,b]上连续的凸函数,我们证明了Hadamard的不等式
$[f(\frac{{a + b}}{2}) \le \frac{1}{{b - a}}\int_a^b {f(x)dx \le \frac{{f(a) + f(b)}}{2}}$
可以拓广成对[a,b]中任意n+1个点x_0,\cdots,x_n和正数组p_0,\cdots,p_n都成立的下列不等式
$f(\frac{\sum\limits_{i=0}^n p_ix_i}{\sum\limits_{i=0}^n p_i}) \leq |\Omega|^-1 \int_\Omega f(x(t))dt \leq \frac{\sum\limits _{i=0}^n {p_if(x_i)}}{\sum\limits_{i=0}^n p_i}$
式中\Omega是一个包含于n维单位立方体的n维长方体,其重心的第i个坐标为$\sum\limits _{j=i}^n p_j /\sum\limits_{j=i-1}^n p_i$,|\Omega|为\Omega的体积,对\Omega中的任意点$t=(t_1,\cdots,t_n)$,
$w(t)=x_0(1-t_1)+\sum\limits _{i=1}^{n-1} x_i(1-t_{i+1})\prod\limits_{j = 1}^i {{t_j}} +x_n \prod\limits _{j=1}^n t_j$
不等式中两个等号分别成立的情形亦已被分离出来。
此不等式是著名的Jensen 不等式的精密化。 相似文献
11.
Oto Strauch 《Monatshefte für Mathematik》1995,120(2):153-164
It is shown that the following three limits
相似文献
12.
Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X
1,X
2, … is any sequence of integrable i.i.d. random variables, then
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