We consider a complex symmetric sequence space E that possesses the Fatou property and is different from l2. We prove that, for every surjective linear isometry V on E, there exist λn ∈ ? with |λn| = 1 and a bijective mapping π on the set ? of natural numbers such that
Let {Xn}n?≥?1 be a sequence of strictly stationary m-dependent random variableswith EX1 = 0 and \( \mathrm{E}{X}_1^2<\infty \), and let (bn) be an increasing sequence of positive numbers such that bn?↑?∞ and \( {b}_n/\sqrt{n}\downarrow 0\kern0.5em \mathrm{as}\kern0.5em n\to \infty \). We establish a moderate deviation principle of \( {\left({b}_n\sqrt{n}\right)}^{-1}{\sum}_{i=1}^n{X}_i \) under the condition
which is weaker than the classical exponential integrability condition. The results in the present paper weaken the assumptions of Chen [5] and extend partially the results of Eichelsbacher and Löwe [10]. 相似文献
For any x ?? (0, 1], let the series \( {\sum}_{n=1}^{\infty }1/{d}_n(x) \) be the Sylvester expansion of x, where {dj(x),?j?≥?1} is a sequence of positive integers satisfying d1(x)?≥?2 and dj?+?1(x)?≥?dj(x)(dj(x)???1)?+?1 for j?≥?1. Suppose ? : ? → ?+ is a function satisfying ?(n+1) – ? (n) → ∞ as n → ∞. In this paper, we consider the set
and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set \( \left\{x\in \kern1em \left(0,1\right]:\kern0.5em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{n}^{\beta }=\upgamma \right\}, \) and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set \( \left\{x\kern0.5em \in \kern0.5em \left(0,1\right]:\kern1em {\lim}_{n\to \infty}\left(\log {d}_n(x)-{\sum}_{j=1}^{n-1}\log {d}_j(x)\right)/{\tau}^n=\eta \right\}. \)相似文献
In this paper we obtain a necessary and sufficient condition on the sequence of natural numbers {qn} such that the almost everywhere convergence of the cubic partial sums Sqn(x) of the multiple Haar series Σnanχn(x) and the condition lim inf \(\lambda \cdot mes\left\{ {x:\begin{array}{*{20}{c}} {\sup } \\ n \end{array}\left| {S{}_{qn}\left( x \right)} \right| \succ \lambda } \right\} = 0\), imply that the coefficients an can be uniquely determined by the sum of the series. Also, we have obtained a necessary and sufficient condition for the series \(\sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}} {\chi _n}\left( x \right)\) with an arbitrary bounded sequence {εn} to be a Fourier-Haar series of an A-integrable function. 相似文献
The paper considers the series by Haar system \(\sum\limits_{n = 1}^\infty {a_n \chi _n (x)} \), satisfying the conditions \(\sum\limits_{n = 1}^\infty {a_n^2 \chi _n^2 (x)} = \infty \) and anχn(x) → 0 almost everywhere. Some theorems about correcting a function on sets of arbitrarily small measures are proved. 相似文献
The Hartman–Wintner–Strassen law of the iterated logarithm states that if X1, X2,… are independent identically distributed random variables and Sn=X1+???+Xn, then
if and only if EX12=1 and EX1=0. We extend this to the case where the Xn are no longer identically distributed, but rather their distributions come from a finite set of distributions.
Let {Xn; n≥1} be a sequence of independent copies of a real-valued random variable X and set Sn=X1+???+Xn, n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,
$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$
where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {Vn; n≥1} is a sequence of independent Banach-space-valued random variables, and an≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).
in the space of analytic functions on the unit polydisk Un in the complex vector space ?n. We show that the operator is bounded in the mixed norm space
, with p, q ∈ [1, ∞) and α = (α1, …, αn), such that αj > ?1, for every j = 1, …, n, if and only if \(\sup _{z \in U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| < \infty \). Also, we prove that the operator is compact if and only if \(\lim _{z \to \partial U^n } \prod\nolimits_{j = 1}^n {\left( {1 - \left| {z_j } \right|} \right)} \left| {g(z)} \right| = 0\).
Divided differences forf (x, y) for completely irregular spacing of points (xi,yi) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (xi,yi) to be at corners of rectangles, or give polynomials Σajkxjyk having more coefficients than interpolation conditions. Here the generalizednth divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (xi,yi), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inxijyik. The generalization of Newton's div. diff. formula is (5)
Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the ki-Hessian operator, a1, p1, f1, a2, p2 and f2 are continuous functions.
For positive integers n, m, and h, we let \({\rho\hat (\mathbb{Z}_n, m, h)}\) denote the minimum size of the h-fold restricted sumset among all m-subsets of the cyclic group of order n. The value of \({\rho\hat (\mathbb{Z}_n, m, h)}\) was conjectured for prime values of n and h = 2 by Erd?s and Heilbronn in the 1960s; Dias da Silva and Hamidoune proved the conjecture in 1994 and generalized it for an arbitrary h, but little is known about the case when n is composite. Here we exhibit an explicit upper bound for all n, m, and h; our bound is tight for all known cases (including all n, m, and h with \({n \leqq 40}\)). We also provide counterexamples for conjectures made by Plagne and by Hamidoune, Lladó, and Serra. 相似文献
In this paper, the system of exponents \(\{ e^{i\lambda _n t} \} _{n \in Z} \) is considered, where {λn} ? R has the following asymptotic form: λn = n ? α sign n + O(|n|?β). Basis properties of this system in Lebesgue space with variable summability factor are investigated. 相似文献
Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ? and B ∈ L2, let \({{\Bbb S}_{B,n}}\) be the space of functions s of the form \(s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} \). This paper describes all spaces \({{\Bbb S}_{B,n}}\) that satisfy the exact inequality \(E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}\). (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases. 相似文献
The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {An}. It generates a two-dimensional array of approximations \({A^{(j)}_n}\) to the limit or anti-limit of {An} defined as solutions of the linear systems
where \({\bar{\beta}_{k}}\) are additional unknowns. In this work, we study the convergence and stability properties of \({A^{(j)}_n}\) , as j → ∞ with n fixed, derived from general linear sequences {An}, where \({{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}\) as n → ∞, where ζk ≠ 1 are distinct and |ζ1| = ... = |ζm| = θ, and γk ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {An}. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems
have unique (integer) solutions for s1, . . . , sm. A special case of our convergence result concerns the situation in which \({{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}\) and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads \({A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}\) as j → ∞. When compared with Aj ? A = O(θj jα) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.
We obtain the operator norms of the n-dimensional fractional Hardy operator Hα (0 < α < n) from weighted Lebesgue spaces \(L_{\left| x \right|^\rho }^p (\mathbb{R}^n )\) to weighted weak Lebesgue spaces \(L_{\left| x \right|^\beta }^{q,\infty } (\mathbb{R}^n )\). 相似文献
We consider the partial-sum process \( {S}_n(t)={\sum}_{k=0}^{\left\lfloor nt\right\rfloor }{X}_k \) of linear processes \( {X}_n={\sum}_{i=0}^{\infty }{c}_i{\upxi}_{n-i} \) with independent identically distributed innovations {ξi} belonging to the domain of attraction of α-stable law (0 < α ≤ 2). If |ck|?=?k?γ?,?k?∈???,?γ?> max(1, 1/α), and \( {\sum}_{k=0}^{\infty}\kern0.5em ck=0 \) (the case of negative memory for the stationary sequence {Xn}), then it is known that the normalizing sequence of Sn(1) can grow as n1/α?γ+1 or remain bounded if the signs of the coefficients are constant or alternate, respectively. It is of interest to know whether it is possible, given ? ∈ (0, 1/α ? γ + 1), to change the signs of ck so that the rate of growth of the normalizing sequence would be n?. In this paper, we give the positive answer: we propose a way of choosing the signs and investigate the finite-dimensional convergence of appropriately normalized Sn(t) to linear fractional Lévy motion. 相似文献
We show a joint denseness theorem for values of a general Dirichlet series \( \sum\nolimits_{n = 1}^\infty {{{\text{a}}_n}{{\text{e}}^{ - {\lambda_n}s}}} \) in a certain class and its derivatives, where the numbers {λn} are not necessarily linearly independent over \( \mathbb{Q} \). 相似文献
Let Λ={λ1,…,λp} be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system
Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators. 相似文献
Let (S,ω) be a weighted abelian semigroup, let Mω(S) be the semigroup of ω-bounded multipliers of S, and let \(\mathcal {A}\) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of \(\ell ^{1}(S,\omega , \mathcal {A})\) if and only if there exists an invertible σ ∈ Mω(S), a unitary point \(a \in \mathcal {A}\), and a k>0 such that \(T(f)= ka{\sum }_{x \in S} f(x)\delta _{\sigma (x)}\) for each \(f={\sum }_{x \in S}f(x)\delta _{x} \in \ell ^{1}(S,\omega ,\mathcal {A})\). It is also shown that an isomorphism from \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\) onto \(\ell ^{1}(S_{2},\omega _{2}, \mathcal {B})\) induces an isomorphism from \(M(\ell ^{1}(S_{1},\omega _{1},\mathcal {A}))\), the set of all multipliers of \(\ell ^{1}(S_{1},\omega _{1},\mathcal {A})\), onto \(M(\ell ^{1}(S_{2},\omega _{2},\mathcal {B}))\). 相似文献