Products of polynomials in uniform norms

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment . Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.

     

Existence of singular extremals and singular functionals in reachable spaces

Given a linear, infinite dimensional control system with point target and "full" control we show that singular extremals for the minimum norm problem exist except in certain exceptional cases ("singular" means "not satisfying Pontryagin's maximum principle"). Existence of singular extremals implies existence of certain functionals (also called singular) in the space of reachable states. Received March 5, 2001; accepted April 10, 2001.     

齐次群上的限制算子与Riesz平均

主要讨论一类齐次群上限制算子的映照性质,并且应用所得结果证明这类齐次群上Riesz平均的混合范数有界性。     

Criterion on Weighted L~p－Boundedness for Oscillatory Singular Integrals with Rough Kernel

ＣｒｉｔｅｒｉｏｎｏｎＷｅｉｇｈｔｅｄＬ~ｐ－ＢｏｕｎｄｅｄｎｅｓｓｆｏｒＯｓｃｉｌｌａｔｏｒｙＳｉｎｇｕｌａｒＩｎｔｅｇｒａｌｓｗｉｔｈＲｏｕｇｈＫｅｒｎｅｌ￥ＪｉａｎｇＹｉｎｓｈｅｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓｏｆＸｉｎｊｉ?..     

Estimates for the Approximation Numbers of One Class of Integral Operators. II

We obtain estimates for the Schatten–von Neumann norms of the approximation numbers of Hardy integral operator with variable limits of integration.     

Random Linear Combinations of Functions from L 1

In this paper, properties of random polynomials with respect to a general system of functions are studied. Some lower bounds for the mathematical expectation of the uniform norm and the recently introduced integral-uniform norm of random polynomials are established.     

一类极大算子的加权范数不等式

本文利用Fourier变换方法和变测度内插定理研究一类极大算子的加权范数不等式，解决了Watson, D. K.在[[6〕中提出的问题.     

带边界Riemann流形上的Sobolev空间等价范数

对具光滑边界αＭ的Ｒｉｅｍａｎｎ流形（Ｍ，ｇ），本文建立了Ｓｏｂｏｌｅｖ空间Ｈ（Ｍ）的等价范数     

BMO空间中f与f_N的范数比较

ＢＭＯ空间中ｆ与ｆ_Ｎ的范数比较刘为铨（安徽芜湖师专）设ｆ（ｘ）为定义于Ｒ上的局部可积函数，Ｑ为Ｒ中的任一立方体。｜Ｑ｜为Ｑ的Ｌｅｂｅｓｅｕｅ测度，ｆ（Ｘ）在Ｑ上的平均值记作若ｆ（Ｘ）满足条件则称人工）为有界平均振动函数，一切有界平均振动函数所构成的?..