Mathematical Sciences: Hankel Operators and Their Applications

9623231 Peller V.V.Peller将继续研究解析算子函数的超优逼近。这种近似不仅最小化了范数的上确界，而且还最小化了所有进一步奇异值的上确界。特别是，V.V.Peller将研究超优逼近的McMillan次数对初始函数的McMillam次数的依赖关系。目前尚不清楚麦克米兰学位能否跳升。另一个悬而未决的问题是超优逼近算子是否保持Wiener代数。V.V.Peller还将研究Nehari-Takagi问题和其他关于超最佳逼近的问题中所谓的主题分解中的指数。V·V·佩勒还将继续他在预测理论方面的工作。最重要的悬而未决的问题之一是用谱密度来刻画完全正则向量平稳高斯过程。V.V.Peller将把Hankel算子应用于分析、预测理论和控制理论的不同领域。很明显，Hankel算子在应用中扮演着重要的角色。特别是，它们在H-无穷大控制理论中起着决定性的作用。V.V.Peller利用Hankel算子研究了解析矩阵函数的超优逼近。在他与N.J.Young的联合工作中，研究表明，在非常自然的假设下，这样的近似是唯一的，并且可以建设性地找到。V·V·佩勒以及V·V·佩勒与S·R·特雷尔的联合论文进一步发展了该理论。超优逼近在控制理论中起着非常重要的作用。超最优逼近在实际应用中具有重要意义，但仍有许多悬而未决的问题。V·V·佩勒将继续努力。Hankel算子的另一个应用领域是预测理论。V·V·佩勒(V.V.Peller)在这一领域取得了许多强有力的成果(部分是由于与赫鲁晓夫的合作)。在这一领域仍有许多悬而未决的问题。V·V·佩勒希望利用向量汉克尔算子取得进展。

9623231 PELLER V.V. Peller is going to continue to study superoptimal approximation by analytic operator functions. Such approximations minimize not only the supremum of the norms but also the suprema of all further singular values. In particular V.V. Peller is going to study the dependence of the Mcmillan degree of the superoptimal approximant on the Mcmillam degree of the initial function. It is not known whether the Mcmillan degree can jump. Another open problem is whether the operator of superoptimal approximation preserves the Wiener algebra. V.V. Peller is also going to study the indices in the so-called thematic factorizations in the Nehari-Takagi problem and other problem on superoptimal approximation. V.V. Peller is also going to continue his work in prediction theory. One of the most important unsolved problems is to characterize in terms of the spectral densities the completely regular vectorial stationary Gaussian processes. V.V. Peller is going to apply Hankel operators in different domains of analysis, prediction theory and control theory. It has become clear that Hankel operators play a significant role in applications. In particular they play a decisive role in H-infinity control theory. V.V. Peller used Hankel operators to study superoptimal approximations by analytic matrix functions. In his joint work with N.J. Young it was shown that under very natural assumptions such an approximation is unique and can be found constructively. Further development of the theory was given by V.V. Peller and in joint papers of V.V. Peller with S.R. Treil. Superoptimal approximations play a very important role in control theory. There are still many open problems about superoptimal approximations which are very important in applications. V.V. Peller is going to continue to work on them. Another domain of applications of Hankel operators is prediction theory. V.V. Peller obtained many strong results (partly in a joint work with Khrushchev) in this field. The re are still many open problems in this field. V.V. Peller hopes to progress using vectorial Hankel operators.