共查询到20条相似文献,搜索用时 31 毫秒
1.
The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let xi, for i=1, 2, ..., n+2, be defined recursively by xi+1=f(xi), where x1 is an arbitrary real number and f is a polynomial of degree n. Let xi+1?xi≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[xi, ..., xi+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {xi}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation xi+1=f(xi) where x1 is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δn?1=|2n?1/n!an|. Then, if n is even and an<0, there do not exist n + 1 consecutive increments Δxi=xi+1?xi in the solution {xi} with Δxi≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x 1, ...,x n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x j )=x j+1 forj=1, 2, ...,n?1. Definep k (n) to be the least positive integer such that if {1, 2, ...,p k (n)} is partitioned into two sets, then one of the two sets must contain ap k -sequence of lengthn. THEOREM. pn?2(n)≦(n!)(n?2)!/2. 相似文献
2.
H. Brass 《Aequationes Mathematicae》1975,13(1-2):151-154
Remark on the estimation ofE n [x n+2m ]. Let be $$E_n [f]: = \mathop {\inf }\limits_{p \in P_n } \mathop {\sup }\limits_{x \in [ - 1, 1]} |f(x) - p(x)|$$ (P n : set of all polynomials of degreen). Riess-Johnson [4] proved (3) $$E_n [x^{n + 2m} ] = \frac{{n^{m - 1} }}{{2^{n + 2m - 1} (m - 1)!}}[1 + O(n^{ - 1} )],n even.$$ This degree of approximation is realized by expansion in Chebyshev polynomials and by interpolation at Chebyshev nodes. The purpose of this paper is to give a more precise estimation by constructing the polynomial of best approximation on a finite set. This construction is easily done and one obtains the result, that the termO(n ?1) in (3) may be replaced by 1/2(m ? 1) (3m + 2)n ?1 + O(n ?2). 相似文献
3.
Let F n be the nth Fibonacci number. The Fibonomial coefficients \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F\) are defined for n ≥ k > 0 as follows $$\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = \frac{{F_n F_{n - 1} \cdots F_{n - k + 1} }} {{F_1 F_2 \cdots F_k }},$$ with \(\left[ {\begin{array}{*{20}c} n \\ 0 \\ \end{array} } \right]_F = 1\) and \(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_F = 0\) . In this paper, we shall provide several identities among Fibonomial coefficients. In particular, we prove that $$\sum\limits_{j = 0}^{4l + 1} {\operatorname{sgn} (2l - j)\left[ {\begin{array}{*{20}c} {4l + 1} \\ j \\ \end{array} } \right]_F F_{n - j} = \frac{{F_{2l - 1} }} {{F_{4l + 1} }}\left[ {\begin{array}{*{20}c} {4l + 1} \\ {2l} \\ \end{array} } \right]_F F_{n - 4l - 1} ,}$$ holds for all non-negative integers n and l. 相似文献
4.
I. Joó 《Analysis Mathematica》1978,4(1):17-26
Доказываются две тео ремы В которых для класса ортонормированных с истем типа сиг-нум устанавливаются те ж е свойства сходимост и, что и для класса всех ортонорм ированных систем. Теорема 1.Ряд Σс n φ п с з аданными коэффициен тами {с п тогда и талька то гда является безусловно сходящим ся почти всюду для все х ортонормированных с истем типасигнум (опр еделен-ных на (0,1)), когда выполнено у словие $$\mathop \sum \limits_{k = 0}^\infty \left[ {\mathop \sum \limits_{n = 2^{2^k } + 1}^{2^{2^{k + 1} } } (c_n^* )^2 \log ^2 n} \right]^{1/2}< \infty $$ , где с n * — невозрастающ ая перестановка последовательности { сn¦. Для класса равномерн о ограниченных ортонормированных с истем это утверждени е было установлено К. Тандор и [7]. Теорема 2.Если 0<λ 1≦λ 2≦ ... —последовательнос ть чисел, для которой Σλ n ?1 =∞,то существует такая орт онормированная сист ема типа сигнум {φ n } (определе нная на (0,1)), что почти всюду на (0,1) Для ортонормированн ых систем не типа сигн ум это утверждение было док азано в [6] (теорема X). Приведенные результ аты дают ответ на вопр осы, поставленные Тандор и ([6], [7]). 相似文献
5.
И. АгАЕВ 《Analysis Mathematica》1985,11(4):283-301
В РАБОтЕ РАссМАтРИВА УтсьS Р-пОДсИстЕМы О. Н.с. В ЧАстНОстИ, ДОкАжыВА Етсь слЕДУУЩАь тЕОРЕ МА, кОтОРАь НЕУсИльЕМА. тЕОРЕМА.пУсть Р>2 —ЧЕ тНОЕ ЧИслО, δ — пРОИжВО льНОЕ ЧИслО, 0<δ≦p?2,Φ= {Φ n(x)} n=1 N —O.H.C.,x?[0,1],пРИЧЕМ ∥ Φ n∥p≦M, n=1,2,...,N, гДЕР=Р+δ, 0М<∞. тОгДА Иж сИстЕМы Ф МОж НО ВыБРАть пОДсИстЕМ У \(\Phi ' = \left\{ {\varphi _{n_k } } \right\}_{k = 1}^{N'} ,N' \geqq N^{\alpha (\delta )} ,\alpha (\delta ) = \frac{{2\delta }}{{p(p - 2 + \delta )}}\) , тАкУУ, ЧтО Дль лУБОгО п ОлИНОМА \(P(x) = \sum\limits_{k = 1}^{N'} {a_k \varphi _{n_k } (x)} \) ИМЕЕ т МЕстО ОцЕНкА $$(\mathop \sum \limits_{k = 1}^{{\rm N}'} a_k^2 )^{1/2} \leqq \left\| P \right\|_p \leqq c_{p,M,\delta } (\mathop \sum \limits_{k = 1}^{{\rm N}'} a_k^2 )^{1/2} $$ (c p, m, δ — пОстОьННАь, жАВИ сьЩАь тОлькО Отp, M, δ, НО НЕ От N ИлИ кОЁФФИцИЕНтОВ пО лИ-НОМА). пРИВОДьтсь И ДРУгИЕ РЕжУльтАты А НАлОгИЧНОгО хАРАктЕ РА. 相似文献
6.
Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦Xi≦1;i=0, …,n. For 0≦p≦1 denote by ? p n the set of all such martingales satisfying alsoE(X0)=p. Thevariation of a martingale χ 0 n is denoted byV 0 n and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ 0 n ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) . 相似文献
7.
W. Dahmen 《Mathematical Notes》1978,23(5):369-376
For the class Cε={f∈C2π: En, n≤Z+} where \(\left\{ {\varepsilon _n } \right\}_{n \in Z_ + } \) is a sequence of numbers tending monotonically to zero, we establish the following precise (in the sense of order) bounds for the error of approximation by de la Vallée-Poussin sums: (1) $$c_1 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \leqslant \mathop {\sup }\limits_{f \in C_\varepsilon } \left\| {f - V_{n, l} \left( f \right)} \right\|_C \leqslant c_2 \sum\nolimits_{j = n}^{2\left( {n + l} \right)} {\frac{{\varepsilon _j }}{{l + j - n + 1}}} \left( {n \in N} \right)$$ , where c1 and c2 are constants which do not depend on n orl. This solves the problem posed by S. B. Stechkin at the Conference on Approximation Theory (Bonn, 1976) and permits a unified treatment of many earlier results obtained only for special classes Cε of (differentiable) functions. The result (1) substantially refines the estimate (see [1]) (2) $$\left\| {V_{n, l} \left( f \right) - f} \right\|_C = O\left( {\log {n \mathord{\left/ {\vphantom {n {\left( {l + 1} \right) + 1}}} \right. \kern-\nulldelimiterspace} {\left( {l + 1} \right) + 1}}} \right) E_n \left[ f \right] \left( {n \to \infty } \right)$$ and includes as particular cases the estimates of approximations by Fejér sums (see [2]) and by Fourier sums (see [3]). 相似文献
8.
Estimates are obtained for the nonsymmetric deviations Rn [sign x] and Rn [sign x]L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e?n, e?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]). 相似文献
9.
S. B. Stečkin 《Analysis Mathematica》1978,4(1):61-74
Пустьf — непрерывная периодическая функц ия,s n (f) — сумма Фурье порядкаn функцииf,E n (f) — наилучшее прибли жениеf тригонометри ческими полиномами порядкаn в чебьппев-ской метрике и $$\sigma _{n, m} (f) = \frac{1}{{m + 1}}\mathop \sum \limits_{v = n - m}^n s_v (f) (0 \leqq m \leqq n; n = 0, 1, \ldots )$$ — суммы Bалле Пуссена ф ункцииf Для любой последовательностиε={εv} (v=0, l,...),ε v ↓0(v→∞) обозначим чер езC(ε) класс непрерывн ых функцийf, для которыхE v (f)≦ε v (v=0,1,...). В работе устанавли вается, что существую т абсолютные положите льные кон-стантыa 1 иa 2 такие, что $$A_1 \mathop \sum \limits_{v = 0}^n \frac{{\varepsilon _{n - m + v} }}{{m + v + 1}} \leqq \mathop {\sup }\limits_{f \in C(\varepsilon )} \parallel f - \sigma _{n, m} (f)\parallel \leqq A_2 \mathop \sum \limits_{v = 0}^n \frac{{\varepsilon _{n - m + v} }}{{m + v + 1}}$$ для всех 0≦m≦n; n=0, l, ... В частн ых случаяхт=п иm=0 этот результат равноси-ле н теоремам, установлен ным ранее автором и К. И. Осколковым. 相似文献
10.
А. А. Женсыкбаев 《Analysis Mathematica》1979,5(4):301-331
The following quadrature formulae are considered: $$\int\limits_0^1 {f(x)dx = \mathop \sum \limits_{k = 1}^n a_k f(x_k ) + \mathop \sum \limits_{i = 1}^l } b_i f^{(\alpha _i )} (0) + \mathop \sum \limits_{j = 1}^m c_j f^{(\beta _j )} (1) + R(f),$$ where 0≦x12<...n≦1 0≦α i , βj≦r?1;l, m, n, andr are positive integers. The problem of existence and uniqueness of the best quadrature formula is solved for the classesW r L p (r=1, 2, ...; 1<p≦∞), obtaining the characteristic properties of its nodesx k and weightsa k ,b i , andc j . 相似文献
11.
А. А. Пекарский 《Analysis Mathematica》1991,17(2):153-171
The paper deals with the order of best rational approximation of some classes of functions, depending on their differentiability properties. Improvements and generalizations of some results by P. P. Petrushev, V. A. Popov and the author are obtained. The proofs are based on the author's direct rational approximation theorems received recently. One of the results reads as follows. LetR n (f,L p ) denote the value of the best approximation of a functionf inL p ,f∈L p [0,1], by rational fractions of degree not exceedingn, n≧1. Suppose that 0<p≦∞,s∈NU{0}, andp≠∞ fors=0. Iff is thes-th primitive of some function of bounded variation on [0,1], then $$\sum\limits_{n = 1}^\infty {\frac{1}{n}(n^{s + 1} R_n (f,L_p ))^2< \infty } $$ . This statement is exact. Namely, for everys, s∈NU {0}, and every sequence {a n } n=1 ∞ , $$a_n \geqq a_{n + 1} and \sum n^{ - 1} (n^{s + 1} a_n )^2< \infty ,$$ , there exists a functiong of the classC s+1 [0,1] satisfying the inequalities $$R_n (g, L_p ) \geqq c(p)a_{12} , n = 1, 2, \ldots ,$$ , for everyp, p∈(0, ∞). 相似文献
12.
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
13.
P. A. Terekhin 《Mathematical Notes》2008,83(5-6):657-674
We obtain conditions for the convergence in the spaces L p [0, 1], 1 ≤ p < ∞, of biorthogonal series of the form $$ f = \sum\limits_{n = 0}^\infty {(f,\psi _n )\phi _n } $$ in the system {? n } n≥0 of contractions and translations of a function ?. The proposed conditions are stated with regard to the fact that the functions belong to the space $ \mathfrak{L}^p $ of absolutely bundleconvergent Fourier-Haar series with norm $$ \left\| f \right\|_p^ * = \left| {f,\chi _0 } \right| + \sum\limits_{k = 0}^\infty {2^{k({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})} } \left( {\sum\limits_{n = 2^k }^{2^{k + 1} - 1} {\left| {f,\chi _n } \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where (f,χ n ), n = 0, 1, ..., are the Fourier coefficients of a function f ? L p [0, 1] in the Haar system {χ n } n≥0. In particular, we present conditions for the system {? n } n≥0 of contractions and translations of a function ? to be a basis for the spaces L p [0, 1] and $ \mathfrak{L}^p $ . 相似文献
14.
Г. Г. Геворкян 《Analysis Mathematica》1988,14(3):219-251
В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥an∥>0.2. Если b n ¦=о(ε n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b n =0для всех п. Теорема 3.Пусть ? n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:
- Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦an≦¦n¦ для n=±1, ±2, ...
- Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦bv¦=о(ε ¦n¦), то bn=0 для всех я. Теорема 5. Пусть послед овательности S(1)={ε 0 (1) , ε 1 (1) , ε 2 (1) , ...} u S2=ε 0 (2) , ε 1 (2) . ε 2 (2) монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S1), но не U(S1) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].
15.
A. A. Yukhimenko 《Mathematical Notes》2007,81(5-6):695-707
We consider the system of exponentials $e(\Lambda ) = \{ e^{i\lambda _n t} \} _{n \in \mathbb{Z}} $ , where $$\lambda _n = n + \left( {\frac{{1 + \alpha }}{p} + l(\left| n \right|)} \right) sign n,$$ l(t) is a slowly varying function, and l(t) → 0, t → ∞. We obtain an estimate for the generating function of the sequence {λn} and, with its help, find a completeness criterion and a basis condition for the system e(Λ) in the weight spaces L p(?π, π). We also study some special cases of the function l(t). 相似文献
16.
A. I. Podvysotskaya 《Ukrainian Mathematical Journal》2009,61(5):847-853
We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than
( n - 1 ) \mathord