共查询到20条相似文献,搜索用时 62 毫秒
1.
V. A. Rodin 《Analysis Mathematica》1990,16(4):291-302
пУстьS k (f,x) — ЧАстНАь сУ ММА РьДА ФУРьЕ ФУНкцИ Иf пО тРИгОНОМЕтРИЧЕскОИ сИстЕМЕ,s k (f,x) — ЧАстНАь сУММА сО пРьжЕННОгО РьДА. Дль \(\Delta _k^n = \left[ {\frac{n}{n},\frac{{k + 1}}{n}} \right)\) , гДЕk=0, 1, ...,n?1, пОлОжИМ , ЕслИt?δ k n И , ЕслИt?[0, 1)δ k n . пОкАжАНО, ЧтО ОпЕРАтО Ры ИМЕУт слАБыИ тИп (1,1). РАссМОтР ЕН РьД слЕДстВИИ О пОВ ЕДЕНИИ сИльНых сРЕДНИх РьДА ФУРьЕ сУММИРУЕМОИ ФУНкцИИ. 相似文献
2.
Let {α n | n ∞ be a sequence in the open unit disk in the complex plane and let $(\overline {\alpha _k } |\alpha _k | = - 1$ when α k =0. Let μ be a positive Borel measure on the unit circle, and let {φ n } n ∞ be the orthonormal sequence obtained by orthonormalization of the sequence {B n } n ∞ with respect to μ. Let {ψ n } n ∞ be the sequence of associated rational functions. Using the functions φ n , ψ n and certain conjugates of them, we obtain modified Padé-type approximants to the function $$F\mu (z) = \int\limits_{ - \pi }^\pi {\frac{{t + z}}{{t - z}}} d\mu (\theta ), (t = e^{i\theta } ).$$ 相似文献
3.
S. K. Bloshanskaya I. L. Bloshanskii 《Proceedings of the Steklov Institute of Mathematics》2014,285(1):34-55
We obtain a criterion for the validity of weak generalized localization almost everywhere on an arbitrary set of positive measure \(\mathfrak{A}\) , \(\mathfrak{A} \subset \mathbb{I}^N = \{ x \in \mathbb{R}^N :0 \leqslant x_j < 1,j = 1,2, \ldots ,N\}\) , N ≥ 3 (in terms of the structure and geometry of the set \(\mathfrak{A}\) ), for multiple Walsh-Fourier series (summed over rectangles) of functions f in the classes \(L_p (\mathbb{I}^N )\) , p > 1 (i.e., necessary and sufficient conditions for the convergence almost everywhere of the Fourier series on some subset of positive measure \(\mathfrak{A}_1\) of the set \(\mathfrak{A}\) , when the function expanded in a series equals zero on \(\mathfrak{A}\) ), in the case when the rectangular partial sums S n (x; f) of this series have indices n = (n 1, …, n N ) ∈ ? N in which some components are elements of (single) lacunary sequences. 相似文献
4.
Rolf Trautner 《Analysis Mathematica》1984,10(1):43-51
По определению после довательность {μ n пр инадлежит классуG s , если звезда М иттагЛеффлера произвольного степе нного ряда (1) $$\mathop \sum \limits_0^\infty a_n z^n , \mathop {lim sup}\limits_{n \to \infty } \left| {a_n } \right|^{1/n}< \infty $$ , совпадает со звёздам и Миттаг-Леффлера сте пенных рядов $$\mathop \sum \limits_0^\infty \mu _n a_n z^n ,\mathop \sum \limits_0^\infty \mu _n^{ - 1} a_n z^n $$ . В работе установлены следующие утвержден ия Теорема 1.Для произво льной последователь ности ? n с условиями $$0< \varphi _n< 1,\mathop {lim}\limits_{n \to \infty } \varphi _n = 0,\mathop {lim}\limits_{n \to \infty } \varphi _n^{1/n} = 1$$ существует неубываю щая функция χ(t) такая, ч то моменты \(\mu _n = \int\limits_0^1 {t^n d\chi (t)} \) удовлетворяют условию 0<μnn звезда М иттаг-Леффлера любог о ряда (1) совпадает со звездой МиттагЛеффлера степенных рядов . Теорема 2. Для произвол ьной неотрицательно й последовательности {аn} с условием {a n } и для любой последов ательности {?n} для к оторой 0n<1, \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n = 0\) сущест вуютπ={π n }∈G s и последовательнос ть {пi} такие, что anμn≦1 (n≧n0), \(a_{n_i } \mu _{\mu _i } \geqq exp( - \varepsilon _{n_i } )\) (i=1, 2, ...) и при эmom звезда Миттаг-Леффлера ряда (1) совпа дает со звездой Миттаг- Леффлера степенных р ядов . 相似文献
5.
The Taikov functional in the space of algebraic polynomials on the multidimensional Euclidean sphere
M. V. Deikalova 《Mathematical Notes》2008,84(3-4):498-514
We discuss three related extremal problems on the set of algebraic polynomials of given degree n on the unit sphere $ \mathbb{S}^{m - 1} $ of Euclidean space ? m of dimension m ≥ 2. (1) The norm of the functional F(h) = FhP n = ∫?(h) P n (x)dx, which is equal to the integral over the spherical cap ?(h) of angular radius arccos h, ?1 < h < 1, on the set with the norm of the space L( $ \mathbb{S}^{m - 1} $ ) of summable functions on the sphere. (2) The best approximation in L ∞( $ \mathbb{S}^{m - 1} $ ) of the characteristic function χ h of the cap ?(h) by the subspace of functions from L ∞( $ \mathbb{S}^{m - 1} $ ) that are orthogonal to the space of polynomials . (3) The best approximation in the space L( $ \mathbb{S}^{m - 1} $ ) of the function χ h by the space of polynomials . We present the solution of all three problems for the value h = t(n,m) which is the largest root of the polynomial in a single variable of degree n + 1 least deviating from zero in the space L 1 ? on the interval (?1, 1) with ultraspheric weight ?(t) = (1 ? t 2) α , α = (m ? 3)/2. 相似文献
6.
We consider a centered Gaussian random field X = {X t : t ∈ T} with values in a Banach space $\mathbb{B}$ defined on a parametric set T equal to ? m or ? m . It is supposed that the distribution of X t is independent of t. We consider the asymptotic behavior of closed convex hulls W n = conv{X t : t ∈ T n}, where (T n ) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b n ) n≥1 with probability 1, (in the sense of Hausdorff distance), where the limit set is the concentration ellipsoid of . The asymptotic behavior of the mathematical expectations Ef(W n ), where f is some function, is also studied. 相似文献
7.
We prove that determinacy for all Boolean combinations of \({F_{\sigma \delta }}\) (Π 3 0 ) sets implies the consistency of second-order arithmetic and more. Indeed, it is equivalent to the statement saying that for every set X and every number n, there exists a β-model of Π n 1 -comprehension containing X. We prove this result by providing a careful level-by-level analysis of determinacy at the finite level of the difference hierarchy on \({F_{\sigma \delta }}\) (Π 3 0 ) sets in terms of both reverse mathematics, complexity and consistency strength. We show that, for n ≥ 1, determinacy for sets at the nth level in this difference hierarchy lies strictly between (in the reverse mathematical sense of logical implication) the existence of β-models of Π n+2 1 -comprehension containing any given set X, and the existence of β-models of Δ n+2 1 -comprehension containing any given set X. Thus the nth of these determinacy axioms lies strictly between Π n+2 1 -comprehension and Δ n+2 1 -comprehension in terms of consistency strength. The major new technical result on which these proof theoretic ones are based is a complexity theoretic one. The nth determinacy axiom implies closure under the operation taking a set X to the least Σ n+1 admissible containing X (for n = 1; this is due to Welch [9]). 相似文献
8.
Square matrices of the form ${X_n = T_n + f_n(T_n^{-1})^*}$ , where T n is a ${n \times n}$ invertible banded Toeplitz matrix and f n some positive sequence are considered. Convergence via an order estimate is proven for the difference of ${\|X_n^{-1}\|}$ and a function depending only on f n . Fredholmness of the infinite counterpart of T n is shown to greatly affect this result. A correction of a proof in the paper on which the current research is based, is appended as well. 相似文献
9.
Giovanni Matarazzo 《Rendiconti del Circolo Matematico di Palermo》1991,40(2):316-324
We prove the uniqueness of weak solutions of initial boundary value problems Each functiona i is required to be sufficiently smooth and must satisfy the following conditions: \(e) \sum\limits_1^n {ij} \partial _{\eta _j \eta _h }^2 a_i (x, \ldots , \eta 1, \ldots , \eta _n )\xi _i \xi _j \leqslant 0, h = 1, \ldots , n,\) for some positive constantsK 0, α, some non negative constantsK i , some positive functionsH(t)∈L 1(0,T) and for all ξ≡(ξ i ), η≡(η i )∈R n 相似文献
10.
We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A(p 1,..., p n ) built up of atoms p 1,..., p n , BPC \({\vdash}\) A(p 1,..., p n ) if and only if for all arithmetical sentences B 1,..., B n , BA \({\vdash}\) A(B 1,..., B n ). The technique used in our proof can easily be applied to some known extensions of BA. 相似文献
11.
Let λsym2f(n) be the n-th coefficient in the Dirichlet series of the symmetric square L-function associated with a holomorphic primitive cusp form f.We prove Ω± results for λsym2f(n) and evaluate the number of positive(resp.,negative) λsym2f(n) in some intervals. 相似文献
12.
Iskander Aliev 《Discrete and Computational Geometry》2008,39(1-3):59-66
Let L(x)=a 1 x 1+a 2 x 2+???+a n x n , n≥2, be a linear form with integer coefficients a 1,a 2,…,a n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a 1,a 2,…,a n . The main result of this paper asserts that there exist linearly independent vectors x 1,…,x n?1∈? n such that L(x i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a 1,a 2,…,a n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry. 相似文献
13.
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
14.
V. V. Bavula 《Algebras and Representation Theory》2014,17(1):275-288
In contrast to its subalgebra $A_n:=K\langle x_1, \ldots , x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}\rangle $ of polynomial differential operators (i.e. the n’th Weyl algebra), the algebra ${\mathbb{I}}_n:=K\langle x_1, \ldots ,$ $ x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}, \int_1, \ldots , \int_n\rangle $ of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that ${\mathbb{I}}_n$ is a left (right) coherent algebra iff n?=?1; the algebra ${\mathbb{I}}_n$ is a holonomic A n -bimodule of length 3 n and has multiplicity 3 n with respect to the filtration of Bernstein, and all 3 n simple factors of ${\mathbb{I}}_n$ are pairwise non-isomorphic A n -bimodules. The socle length of the A n -bimodule ${\mathbb{I}}_n$ is n?+?1, the socle filtration is found, and the m’th term of the socle filtration has length ${n\choose m}2^{n-m}$ . This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra ${\mathbb{I}}_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra ${\mathbb{I}}_n$ . 相似文献
15.
Norman R. Reilly 《Semigroup Forum》2014,89(1):249-265
The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS n generated by all completely 0-simple semigroups over groups from the Burnside variety G n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B 2 , generated by the five element Brandt semigroup B 2, and contained in the variety NB 2 ∨G n where NB 2 is the variety generated by all left and right zero semigroups together with B 2. The interval [NB 2 ,NB 2 ∨G n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS n . 相似文献
16.
We investigate the boundedness nature of positive solutions of the difference equation $$ x_{n + 1} = max\left\{ {\frac{{A_n }} {{X_n }},\frac{{B_n }} {{X_{n - 2} }}} \right\},n = 0,1,..., $$ where {A n } n=0 ∞ and {B n } n=0 ∞ are periodic sequences of positive real numbers. 相似文献
17.
We study multiple trigonometric Fourier series of functions f in the classes $L_p \left( {\mathbb{T}^N } \right)$ , p > 1, which equal zero on some set $\mathfrak{A}, \mathfrak{A} \subset \mathbb{T}^N , \mu \mathfrak{A} > 0$ (µ is the Lebesgue measure), $\mathbb{T}^N = \left[ { - \pi ,\pi } \right]^N$ , N ≥ 3. We consider the case when rectangular partial sums of the indicated Fourier series S n (x; f) have index n = (n 1, ..., n N ) ∈ ? N , in which k (k ≥ 1) components on the places {j 1, ..., j k } = J k ? {1, ..., N} are elements of (single) lacunary sequences (i.e., we consider multiple Fourier series with J k -lacunary sequence of partial sums). A correlation is found of the number k and location (the “sample” J k ) of lacunary sequences in the index n with the structural and geometric characteristics of the set $\mathfrak{A}$ , which determines possibility of convergence almost everywhere of the considered series on some subset of positive measure $\mathfrak{A}_1$ of the set $\mathfrak{A}$ . 相似文献
18.
Skorokhod's representation theorem states that if on a Polish space,there is a weakly convergent sequence of probability measures μnw→μ0,as n →∞,then there exist a probability space(Ω,F,P) and a sequence of random elements Xnsuch that Xn→ X almost surely and Xnhas the distribution function μn,n = 0,1,2,... We shall extend the Skorokhod representation theorem to the case where if there are a sequence of separable metric spaces Sn,a sequence of probability measures μnand a sequence of measurable mappings n such that μnn-1w→μ0,then there exist a probability space(Ω,F,P) and Sn-valued random elements Xndefined on Ω,with distribution μnand such that n(Xn) → X0 almost surely. In addition,we present several applications of our result including some results in random matrix theory,while the original Skorokhod representation theorem is not applicable. 相似文献
19.
Let x = (x n ) n?1 be a martingale on a noncommutative probability space ( $\mathcal{M}$ , τ) and (w n ) n?1 a sequence of positive numbers such that $W_n = \sum\nolimits_{k = 1}^n {w_k \to \infty } $ as n → ∞. We prove that x = (x n ) n?1 converges bilaterally almost uniformly (b.a.u.) if and only if the weighted average (σ n (x)) n?1 of x converges b.a.u. to the same limit under some condition, where σ n (x) is given by $\sigma _n (x) = \frac{1} {{W_n }}\sum\limits_{k = 1}^n {w_k x_k } ,n = 1,2,... $ Furthermore, we prove that x = (x n ) n?1 converges in L p ( $\mathcal{M}$ ) if and only if (σ n (x)) n?1 converges in L p ( $\mathcal{M}$ ), where 1 ? p < ∞. We also get a criterion of uniform integrability for a family in L 1( $\mathcal{M}$ ). 相似文献
20.
Wei-Bin Zeng 《Journal of Theoretical Probability》1995,8(1):1-15
LetX 1,X 2, ...,X n be independent and identically distributed random vectors inR d , and letY=(Y 1,Y 2, ...,Y n )′ be a random coefficient vector inR n , independent ofX j /′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY. 相似文献