• Korean Journal of Mathematics
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• 제30권2호
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• pp.305-313
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• 2022

In this paper we consider a modified version of Smirnov operator and obtain some Bernstein-type inequalities preserved by this operator. In particular, we prove some results which in turn provide the compact generalizations of some well-known inequalities for polynomials.

• Nonlinear Functional Analysis and Applications
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• 제28권1호
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• pp.287-294
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• 2023

The goal of this paper is to extend some inequalities of Bernstein as well as Turán type to polar derivative of a polynomial.

• 대한수학회보
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• 제36권1호
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• pp.63-78
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• 1999

Let $\mu$(x) be an increasing function on the real line with finite moments of all oeders. We show that for any linear operator T on the space of polynomials and any interger n $\geq$ 0, there is a constant $\gamma n(T)\geq0$, independent of p(x), such that $\parallel T_p\parallel\leq\gamma n(T)\parallel P\parallel$, for any polynomial p(x) of degree $\leq$ n, where We find a formular for the best possible value $\Gamma_n(T)\;of\;\gamma n(T)$ and estimations for $\Gamma_n(T)$. We also give several illustrating examples when T is a differentiation or a difference operator and $d\mu$(x) is an orthogonalizing measure for classical or discrete orthogonal polynomials.

• Communications for Statistical Applications and Methods
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• 제21권3호
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• pp.245-251
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• 2014

Two exponential inequalities are established for a wide class of general weakly dependent sequences of random variables, called ${\psi}$-weakly dependent process which unify weak dependence conditions such as mixing, association, Gaussian sequences and Bernoulli shifts. The ${\psi}$-weakly dependent process includes, for examples, stationary ARMA processes, bilinear processes, and threshold autoregressive processes, and includes essentially all classes of weakly dependent stationary processes of interest in statistics under natural conditions on the process parameters. The two exponential inequalities are established on more general conditions than some existing ones, and are proven in simpler ways.

• KSII Transactions on Internet and Information Systems (TIIS)
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• 제12권1호
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• pp.21-40
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• 2018

In this paper, we propose the distributed robust beamforming design scheme in cognitive two-way amplify-and-forward (AF) relay networks with imperfect channel state information (CSI). Assuming the CSI errors follow a complex Gaussian distribution, the objective of this paper is to design the robust beamformer which minimizes the total transmit power of the collaborative relays. This design will guarantee the outage probability of signal-to-interference-plus-noise ratio (SINR) beyond a target level at each secondary user (SU), and satisfies the outage probability of interference generated on the primary user (PU) above the predetermined maximum tolerable interference power. Due to the multiple CSI uncertainties in the two-way transmission, the probabilistic constrained optimization problem is intractable and difficult to obtain a closed-form solution. To deal with this, we reformulate the problem to the standard form through a series of matrix transformations. We then accomplish the problem by using the probabilistic approach based on two sorts of Bernstein-type inequalities and the worst-case approach based on S-Procedure. The simulation results indicate that the robust beamforming designs based on the probabilistic method and the worst-case method are both robust to the CSI errors. Meanwhile, the probabilistic method can provide higher feasibility rate and consumes less power.