共查询到20条相似文献,搜索用时 15 毫秒
1.
Zamira Abdikalikova Ryskul Oinarov Lars-Erik Persson 《Czechoslovak Mathematical Journal》2011,61(1):7-26
We consider a new Sobolev type function space called the space with multiweighted derivatives $
W_{p,\bar \alpha }^n
$
W_{p,\bar \alpha }^n
, where $
\bar \alpha
$
\bar \alpha
= (α
0, α
1,…, α
n
), α
i
∈ ℝ, i = 0, 1,…, n, and $
\left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|}
$
\left\| f \right\|W_{p,\bar \alpha }^n = \left\| {D_{\bar \alpha }^n f} \right\|_p + \sum\limits_{i = 0}^{n - 1} {\left| {D_{\bar \alpha }^i f(1)} \right|}
,
$
D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d}
{{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n
$
D_{\bar \alpha }^0 f(t) = t^{\alpha _0 } f(t),D_{\bar \alpha }^i f(t) = t^{\alpha _i } \frac{d}
{{dt}}D_{\bar \alpha }^{i - 1} f(t),i = 1,2,...,n
相似文献
2.
Let A
0, ... , A
n−1 be operators on a separable complex Hilbert space , and let α0,..., α
n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,
3.
Let {Xk} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s
Fn(x) of sums
is studied, where 0<α≤2, ank>0, 1≤k≤mn, and, as n→∞, bothmax
1≤k≤mna
nk→0 and
. It is shown that such convergence, with suitably chosen An's and necessarily stable limit laws, holds for all such arrays {αnk} provided it holds for the special case αnk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence
of the moments of the sequence {Fn(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums
, where an≠0, bn>0, andmax
1≤k≤n ak/bn→0. The strong law is said to hold if there are constants An for which Sn→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which bn/|an|<x if x>0. The main result is as follows. If the strong law holds,EN (|X1|)<∞. If
for some 0<p≤2, then the strong law holds with
if 1≤p≤2 and An=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various
conditions imposed on F(x), the coefficients an and bn, and the function N(x).
Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993. 相似文献
4.
Jiang Chaowei Yang Xiaorong 《高校应用数学学报(英文版)》2007,22(1):87-94
In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established. 相似文献
5.
Considering the positive d-dimensional lattice point Z
+
d
(d ≥ 2) with partial ordering ≤, let {X
k: k ∈ Z
+
d
} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $
S_n = \sum\limits_{k \leqslant n} {X_k }
$
S_n = \sum\limits_{k \leqslant n} {X_k }
, n ∈ Z
+
d
. Let σ
i
2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ
2. Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
$
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
. We show that when l ≥ 2 and b > −l/2, E[‖X‖2(log ‖X‖)
d−2(log log ‖X‖)
b+4] < ∞ implies $
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
$
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
, where Γ(·) is the Gamma function and $
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
$
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
. 相似文献
6.
In this paper we prove that if
is a set of
k positive integers and
{A
1,
..., A
m
} is a family of subsets
of an n-element set
satisfying
, for all 1
i <
j m, then
. The case
k = 1 was proven 50 years ago
by Majumdar. 相似文献
7.
Let T = T(p, q, α) be the number of solutions of the congruence xα ≡ 1 (mod pηqθ). Let A
and B be sets of primes satisfying x1 < p ≤ x2 and y1 < q ≤ y2, respectively. A mean value estimation
of
is given.
Supported by National Natural Science Foundation of China (No. 19971024) and Zhejiang Provincial Natural
Science Foundation of China (No. 199047) 相似文献
8.
A. V. Harutyunyan W. Lusky 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2010,45(3):128-135
Let U
n
be the unit polydisk in C
n
and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω
1, ..., ω
n
), ω
j
∈ S(1 ≤ j ≤ n) and f ∈ H(U
n
). The function f is said to be in holomorphic Besov space B
p
(ω) if
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