The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies ...The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.展开更多
The mean-square exponential stability problem is investigated for a class of stochastic time-varying delay systems with Markovian jumping parameters. By decomposing the delay interval into multiple equidistant subinte...The mean-square exponential stability problem is investigated for a class of stochastic time-varying delay systems with Markovian jumping parameters. By decomposing the delay interval into multiple equidistant subintervals, a new delay-dependent and decay-rate-dependent criterion is presented based on constructing a novel Lyapunov functional and employing stochastic analysis technique. Besides, the decay rate has no conventional constraint and can be selected according to different practical conditions. Finally, two numerical examples are provided to show that the obtained result has less conservatism than some existing ones in the literature.展开更多
The goal of this paper is to establish the relations between general Bernstein and Nikol＇skii type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a ...The goal of this paper is to establish the relations between general Bernstein and Nikol＇skii type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a family of functions equipped with Bernstein type inequality which satisfies Nikol＇skii type inequality is found.展开更多
The author considers the L^p boundedness for two kinds of Carleson-type maximal operators with variable kernels(Ω(x,y'))/(|y|~n),whereΩ(x,y')∈L~∞(R^n)×W_2~s(S^(n-1))for some s>0.
文摘The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.
基金supported by the Program for New Century Excellent Talents in University, the Graduate Innovation Program of Jiangsu Province (CX06B-051Z)the Scientific Research Foundation of Graduate School of Southeast University (YBJJ0929)
文摘The mean-square exponential stability problem is investigated for a class of stochastic time-varying delay systems with Markovian jumping parameters. By decomposing the delay interval into multiple equidistant subintervals, a new delay-dependent and decay-rate-dependent criterion is presented based on constructing a novel Lyapunov functional and employing stochastic analysis technique. Besides, the decay rate has no conventional constraint and can be selected according to different practical conditions. Finally, two numerical examples are provided to show that the obtained result has less conservatism than some existing ones in the literature.
基金Supported by the National Science Foundation of China(Grant 90818020, 60874029)
文摘The goal of this paper is to establish the relations between general Bernstein and Nikol＇skii type inequalities under some weak conditions. From these relations some known classical inequalities are implied. Also, a family of functions equipped with Bernstein type inequality which satisfies Nikol＇skii type inequality is found.
基金supported by the National Natural Science Foundation of China(Nos.11226103,11226102)the Doctor Foundation of Henan Polytechnic University(No.B2011-034)
文摘The author considers the L^p boundedness for two kinds of Carleson-type maximal operators with variable kernels(Ω(x,y'))/(|y|~n),whereΩ(x,y')∈L~∞(R^n)×W_2~s(S^(n-1))for some s>0.
