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V. P. Zastavnyi 《Mathematical Notes》2004,75(1-2):175-189
We consider entire functions of exponential type ≤ σ that are bounded and real on $\mathbb{R}$ and satisfy the estimate $( - 1)^k f(k\pi /\sigma + \tau ) \geqslant 0, k \in \mathbb{Z}$ . It is proved that the zeros of such functions are real and simple with the possible exception of points of the form $k\pi /\sigma + \tau $ , which can be zeros of multiplicity at most 2. These results are applied to specific classes of functions and to the problem of the stability of entire functions. We also refine and supplement a few results due to Pólya. 相似文献
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V. P. Zastavnyi 《Ukrainian Mathematical Journal》2003,55(7):1189-1197
We establish sufficient conditions for the extension of a function defined on [a, +), where a > 0, to a positive-definite function on the entire axis. 相似文献
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For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant. 相似文献
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We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A k ? ? and a sequence of functions λ k : A k → ? provide the existence of a number C such that any function f ∈ L 1 satisfies the inequality ‖U A,Λ(f)‖ p ≤ C‖f‖1 and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c m (f) are Fourier coefficients of the function f ∈ L 1. Problem 2: what conditions on a sequence of finite subsets A k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L p for every function h of bounded variation? 相似文献
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Asymptotics of series arising from the approximation of periodic functions by Riesz and Cesàro means
V. P. Zastavnyi 《Mathematical Notes》2013,93(1-2):58-68
Asymptotic expansions in powers of δ as δ → +∞ of the series $\sum\limits_{k = 0}^\infty {( - 1)^{(\beta + 1)k} \frac{{Q((\delta ^\alpha - (ak + b)^\alpha ) + )}} {{(ak + b)^{r + 1} }}} , $ where β ∈ ?, α, a, b > 0, and r ∈ ?, while Q is an algebraic polynomial satisfying the condition Q(0) = 0, are obtained. In special cases, these series arise from the approximation of periodic differentiable functions by the Riesz and Cesàro means. 相似文献
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A parametric family of operators G ρ is constructed for the class of convolutions W p,m (K) whose kernel K was generated by the moment sequence. We obtain a formula for evaluating $E(W_{p,m} (K);G_\rho )_p : = \mathop {\sup }\limits_{f \in W_{p,m} (K)} \left\| {f - G_\rho (f)} \right\|_p .$ . For the case in which W p,m (K)=W r,β p,m , we obtain an expansion in powers of the parameter ?=?ln ρ for E(W p,m r,β ; G ρ,r ) p , where β ∈ ?, γ > 0, and m ∈ ?, while p = 1 or p = ∞. 相似文献
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Mathematical Notes - Given α ∈ (0, 1) and c = h + iβ, h, β ∈ R, the function fα,c: R → C defined as follows is considered: (1) fα,c is Hermitian, i.e.,... 相似文献
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A characterization of bounded sets in Banach spaces in terms of asymptotic cones and the Hausdorff deviations of sets from them homothetic images is obtained. Similar results for generalizations of the notion of boundedness are presented. Boundedness criteria have previously been known only for recessively compact sets.
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