共查询到20条相似文献,搜索用时 15 毫秒
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Ferenc Móricz 《Acta Mathematica Hungarica》2008,121(1-2):1-19
We consider N-multiple trigonometric series whose complex coefficients c j1,...,j N , (j 1,...,j N ) ∈ ? N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α1,..., α N ) and lip (α1,..., α N ) for some α1,..., α N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too. 相似文献
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R. R. Ashurov 《Mathematical Notes》1989,46(4):755-757
Translated from Matematicheskie Zametki, Vol. 46, No. 4, pp. 3–7, October, 1989. 相似文献
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Л. А. Шерстнева 《Analysis Mathematica》1988,14(4):323-345
Quasi-normed Lorentz spaces Λψ, q of 2π-periodic functions with quasinorms $$\left\| f \right\|_{\psi ,q} = \left\{ {\int\limits_0^{2\pi } {\psi ^q (t)\left[ {\frac{1}{t}\int\limits_0^t {f * (x)} dx} \right]} ^q \frac{{dt}}{t}} \right\}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} $$ (0<q<∞,ω(t): [0,2π]→R is a continuous concave function with finite derivative everywhere on (0, 2gp)) and classes of functions $$H_{\psi ,q}^\omega \equiv \{ f(x):f(x) \in \Lambda _{\psi ,q} ;\mathop {\sup }\limits_{0 \leqq h \leqq \delta } \left\| {f(x + h) - f(x)} \right\|_{\psi ,q} = O\{ \omega (\delta )\} , \delta \to + 0\} $$ (ω(δ) — modulus of continuity) are studied. Precise embedding conditions of classes H ψ, q ω into Lorentz spaces and into each other are obtained: $$\begin{array}{*{20}c} {H_{\psi ,q_1 }^\omega \subset \Lambda _{\psi ,q_2 } ;} & {H_{\psi ,q_1 }^\omega \subset {\rm H}_{\psi ,q_2 }^{\omega * } ,} & {0< q_2< q_1< \infty ,} \\ \end{array} $$ under conditions \(\mathop {\lim }\limits_{t \to \infty } \frac{{\psi (2t)}}{{\psi (t)}} > 1,\mathop {\overline {\lim } }\limits_{x \to \infty } \frac{{\psi (2t)}}{{\psi (t)}}< 2\) andω(δ)=O{ω(δ 2)},δ→+0, andω * (δ) is an arbitrary modulus of continuity. 相似文献
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A. A. Dovgoshei 《Ukrainian Mathematical Journal》1992,44(2):159-164
The approximation of functions from Hardy classes by bounded analytic functions is investigated. A theorem is proved, characterizing the sets of functions with equiabsolutely continuous integrals as limit points of the family of bounded subsets of the space H.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 178–184, February, 1992. 相似文献
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N. L. Pachulia 《Ukrainian Mathematical Journal》1989,41(3):313-318
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 354–360, March, 1989. 相似文献
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Shin-ya Koyama Nobushige Kurokawa 《Proceedings of the American Mathematical Society》2005,133(5):1257-1265
We show that Euler's famous integrals whose integrands contain the logarithm of the sine function are expressed via multiple sine functions.
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A. M. Stokolos 《Mathematical Notes》1994,55(1):57-70
Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 84–104, January, 1994. 相似文献
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Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If n denotes the symmetric group on {1,2,…,n} then we define the projection by the formula , where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If , then x1?x2? … ?xn denotes the member of Tn(V) such that for each y1 ,2,…,yn in V, and x1·x2… xn denotes . If B? Sn(V) and there exists , such that B = x1·x2…xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C. 相似文献
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V. L. Grona 《Ukrainian Mathematical Journal》1993,45(10):1490-1505
We investigate the conditions for the localization of the Bochner-Riesz means in the Nikol'skii classesu
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forp[1, 2].Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1331–1344, October, 1993. 相似文献
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Mathematical Notes - We introduce the notion of spherical jump of a function of several variables at a given point with respect to a homogeneous harmonic polynomial. Here, if the function is... 相似文献
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We establish asymptotic equalities for upper bounds of approximations by Fourier sums and for the best approximations in the
metrics of C and L1 on classes of convolutions of periodic functions that can be regularly extended into a fixed strip of the complex plane. 相似文献
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We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the metric of the space of quasi-continuous functions . We also showed that for , , , , the estimate of the corresponding asymptotic characteristic is exact in order. 相似文献
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Krzysztof Ciesielski 《Proceedings of the American Mathematical Society》1999,127(12):3615-3622
In this paper we will investigate the smallest cardinal number such that for any symmetrically continuous function there is a partition of such that every restriction is continuous. The similar numbers for the classes of Sierpinski-Zygmund functions and all functions from to are also investigated and it is proved that all these numbers are equal. We also show that and that it is consistent with ZFC that each of these inequalities is strict.