共查询到20条相似文献,搜索用时 62 毫秒
1.
2.
N. Patrsvania 《Georgian Mathematical Journal》1996,3(6):571-582
Sufficient conditions are found for the oscillation of proper solutions of the system of differential equations $$\begin{array}{*{20}c} {u'_1 (t) = f_1 (t,u_1 (\tau _1 (t)),...,u_1 (\tau _m (t)),u_2 (\sigma _1 (t)),...,u_2 (\sigma _m (t))),} \\ {u'_2 (t) = f_2 (t,u_1 (\tau _1 (t)),...,u_1 (\tau _m (t)),u_2 (\sigma _1 (t)),...,u_2 (\sigma _m (t))),} \\ \end{array}$$ wheref i: R+×R2m→R (i=1,2) satisfy the local Carathéodory conditions andσ i , τ i :R + →R (i=1,...,m) are continuous functions such that $\sigma _i (t) \leqslant t for t \in R_ + ,\mathop {\lim }\limits_{t \to + \infty } \sigma _i (t) = + \infty ,\mathop {\lim }\limits_{t \to + \infty } \tau _i (t) = + \infty (i = 1,...,m)$ 相似文献
3.
Let φ be a supermultiplicative Orlicz function such that the function $t \mapsto \varphi \left( {\sqrt t } \right)$ is equivalent to a convex function. Then each complexn×n matrixT=(τ ij ) i, j satisfies the following eigenvalue estimate: $\left\| {\left( {\lambda _i \left( T \right)} \right)_{i = 1}^n } \right\|_{\ell _\varphi } \leqslant C\left\| ( \right\|\left( {\tau _{ij} } \right)_{i = 1}^n \left\| {_{_{\ell _{\varphi *} } } )_{j = 1}^n } \right\|\ell _{\bar \varphi } $ . Here, ?* stands for Young’s conjugate function of φ, ?, $\bar \varphi $ is the minimal submultiplicative function dominating φ andC>0 a constant depending only on φ. For the power function φ(t)=t p ,p≥2 this is a celebrated result of Johnson, König, Maurey and Retherford from 1979. In this paper we prove the above result within a more general theory of related estimates. 相似文献
4.
5.
Prof. Dr. Ulrich Krengel 《Monatshefte für Mathematik》1978,86(1):3-6
Given any ergodic invertible measure preserving transformation τ of [0,1] and any null-sequence (α N ) of positive reals, there exists a continuousf such that $$\lim \sup \alpha _{\rm N}^{ - 1} \left| {N^{ - 1} \sum\limits_{k = 0}^{N - 1} {f \circ \tau ^k - \smallint f} } \right| = \infty a. e.,$$ i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence. 相似文献
6.
This paper is a continuation of [3]. Suppose f∈Hp(T), 0σ r σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums Skf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ . 相似文献
7.
Adam Osękowski 《Israel Journal of Mathematics》2012,192(1):429-448
The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.
- Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
- Let P + and P ? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
- We establish the sharp versions of the estimates above in the nonperiodic case.
8.
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
9.
S. Ja. Havinson 《Analysis Mathematica》1983,9(2):99-111
Для заданной на едини чной окружности огра ниченной функцииω(ξ) рассматр ивается усложненная задача а ппроксимации аналит ическими функциями: $$\mathop {\inf }\limits_{\varphi \in H^\infty } \left[ {\left\| {\omega - \varphi } \right\| + \mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k \left| {\lambda _k } \right|} \right],$$ где ∥·∥ понимается вL ∞,ε k ≧0 — заданные чис ла, $$\mathop \Sigma \limits_{k = 0}^\infty \varepsilon _k< + \infty ,\varphi (z) = \mathop \Sigma \limits_{k = 0}^\infty \lambda _k z^k .$$ Доказывается, что при всех достаточно малы хε k экстремальной в этой задаче будет функция обычного наилучшего приближения (та же, что и приε k =0,k=0, 1, ...). В частности, при $$\omega (\zeta ) = \frac{{\gamma _0 }}{{\zeta ^n }} + \frac{{\gamma _1 }}{{\zeta ^{n - 1} }} + ... + \frac{{\gamma _{n - 1} }}{\zeta }$$ экстремальной оказы вается дробь Каратео дори—Фейера. Переход к двойственн ой задаче позволяет получить т очные оценки для клас са интегралов типа Коши, выделяемого огранич ениями, наложенными на велич ины коэффициентов ря да Тейлора. 相似文献
10.
E. Quak 《Analysis Mathematica》1988,14(3):259-272
ПустьМ[а, b] — множество вещественных функци йf, измеримых и ограниче нных на отрезке [а, b], иL: M[a, b] → M[a, b] — лин ейный положительный оператор. В статье пол учены оценки погрешностей аппроксимации вL p -нор ме, выраженные в термина х так называемого усредненного локаль ного модуля гладкост и, илиτ-модуля. ПустьL облад ает свойствами: (e i ,i(t):=t i дляi=0,1,2)Le 0 =e 0 ,Le 1 =e 1 ,Le 2 (х)=х 2 +β(х), x∈[a,b] и i)A ≦ min (1, (b-a)2/4) для 1 <p<∞, ii)A ≦ min (e ?1 , (b — a)2) дляр=1 приA:=∥β∥t8. Тогда выполнены оцен ки: $$\begin{gathered} \left\| {f - Lf} \right\|_p \leqq C\frac{p}{{p - 1}}\tau _2 (f;\sqrt A )_p 1< p< \infty \hfill \\ \left\| {f - Lf} \right\|_1 \leqq C\tau _2 \left( {f;\sqrt {A\ln \frac{1}{A}} } \right)_1 , \hfill \\ \end{gathered} $$ где положительные по стоянныеС не завися т от оператораL, функцииfи параметрар. 相似文献
11.
Deh-phone Kung Hsing 《Annali di Matematica Pura ed Applicata》1976,109(1):235-245
Summary We consider the system(L):
, t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each Ai, B(t) are n × n matrices. System(L) generates a semigroup by means of Ttf(s)=y (t+s, f), f(s) ∈ BCl(− ∞, 0]. Under some hypotheses concerning the roots ofdet
where
is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose
∥B(t)∥ ɛ L1[0, ∞),
thendet
forRe λ>0 iff for every ɛ>0 there is an Mɛ>0 such that ∥Ttf∥l ⩽ ⩽ Mɛ
exp [ɛt]∥f∥l for t ⩾ 0. Corollary 3.1.1: suppose
exp [at]B(t) ∈ ∈ L1[0, ∞) for some a>0 anddet
forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable.
Entrata in Redazione il 21 marzo 1975.
The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper. 相似文献
12.
Let {ξi,-∞i∞} be a doubly infinite sequence of identically distributed-mixing random variables with zero means and finite variances,{ai,-∞i∞} be an absolutely summable sequence of real numbers and X k =∑i=-∞+∞ aiξi+k be a moving average process.Under some proper moment conditions,the precise asymptotics are established for 相似文献
13.
V. I. Bernik 《Mathematical Notes》1972,11(6):378-380
The problem of finding the asymptotic number of solutions of the system of inequalities $$\begin{gathered} \left\| {\alpha _i q} \right\|< q^{ - \sigma _i } (i = 1,...,n), \sigma _i > 0, \hfill \\ \sigma = \sum\nolimits_{i = 1}^n {\sigma _i< c(\alpha _1 ,...,\alpha _n ), q = 1,...,N,} \hfill \\ \end{gathered}$$ is solved under the assumption that for real numbers α1,..., αn, starting from some Q=max(q1...,qn) the inequality holds for any real λ≥0. 相似文献
14.
Carsten Schütt 《Israel Journal of Mathematics》1981,40(2):97-117
Suppose{e i} i=1 n and{f i} i=1 n are symmetric bases of the Banach spacesE andF. Letd(E,F)≦C andd(E,l n 2 )≧n' for somer>0. Then there is a constantC r=Cr(C)>0 such that for alla i∈Ri=1,...,n $$C_r^{ - 1} \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\| \leqq \left\| {\sum\limits_{i = 1}^n {a_i f_i } } \right\| \leqq C_r \left\| {\sum\limits_{i = 1}^n {a_i e_i } } \right\|$$ We also give a partial uniqueness of unconditional bases under more restrictive conditions. 相似文献
15.
Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes 总被引:1,自引:0,他引:1
Yun Xia LI Li Xin ZHANG 《数学学报(英文版)》2006,22(1):143-156
In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai+kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞) is an absolutely solutely summable sequence of real numbers. 相似文献
16.
Suppose f∈Hp(Tn), 0 r δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1. 相似文献
17.
Yu. I. Alimov 《Mathematical Notes》1970,8(2):558-563
An investigation of measurable almost-everywhere finite functions ξ(t), -∞
18.
We show that there do not exist computable functions f 1(e, i), f 2(e, i), g 1(e, i), g 2(e, i) such that for all e, i ∈ ω,
19.
В РАБОтЕ пРИВЕДЕНы НЕ ОБхОДИМыЕ И ДОстАтОЧ НыЕ УслОВИь сУЩЕстВОВАНИь НЕРАВ ЕНстВА НА пОлУпРьМОИ R+=[0, ∞): $$\left\| {(D^\alpha x)( \cdot )} \right\|_{C(R_ + )} \leqq K\left\| {x( \cdot )} \right\|_{L_2 (R_ + )}^{v_1 } \left\| {(D^n x)( \cdot )} \right\|_{L_2 (R_ + )}^{v_2 } ,$$ гДЕ А-пРОИжВОльНОЕ ВЕ ЩЕстВЕННОЕ ЧИслО,n≧1 — цЕлОЕ Иv i>0,i=1,2. ДРОБНАь пРОИжВОД НАьD α пОНИМАЕтсь В сМыслЕ г. ВЕИль. ВыЧИслЕНА НАИ лУЧшАь (т.Е. НАИМЕНьшАь Иж ВОжМ ОжНых) кОНстАНтАк=к(п, А) В ЁтО М НЕРАВЕНстВЕ И ВыпИс АНА ЁкстРЕМАльНАь ФУНкц Иь, НА кОтОРОИ НЕРАВЕНстВО пРЕВРАЩАЕтсь В РАВЕН стВО. 相似文献
20.
Jay Taylor 《Israel Journal of Mathematics》2017,217(1):435-475
In this paper we establish the following estimate: 相似文献
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$ $${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$ $$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$ |