论文标题
泰勒系列的统一和建设性分析延续
Uniformization and Constructive Analytic Continuation of Taylor Series
论文作者
论文摘要
我们根据某个时候以截断功率序列的形式和其他分析属性，根据部分信息和其他分析性能分析了尽可能准确的全局功能重建问题。这种情况在应用中经常发生。最佳过程的问题是开放的，我们将其作为一个良好的数学问题提出。它的解决方案导致了一种实用方法，可对现有技术进行巨大的准确性改进。我们的程序基于黎曼表面的均匀化。作为一个应用程序，我们表明我们的过程可以用于一系列非线性ODE的解决方案。我们找到了一种新的统一方法，我们用来构建特殊功能所需的统一地图，包括painlev'e方程式P_I-P_V的解决方案。我们还引入了一种新的严格和建设性的正规化方法，消除了其位置和类型的奇异性。如果这些未知，则相同的过程可以实现一种高度敏感的共振方法来确定奇异性的位置和类型。在有关Riemann Surface的明确信息较少的应用程序中，我们的方法和技术会导致新的近似值，但仍然比现有方法更精确的重建方法，尤其是在奇异之外，这是最引起最大兴趣的点。
We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point, and additional analyticity properties. This situation occurs frequently in applications. The question of the optimal procedure was open, and we formulate it as a well-posed mathematical problem. Its solution leads to a practical method which provides dramatic accuracy improvements over existing techniques. Our procedure is based on uniformization of Riemann surfaces. As an application, we show that our procedure can be implemented for solutions of a wide class of nonlinear ODEs. We find a new uniformization method, which we use to construct the uniformizing maps needed for special functions, including solution of the Painlev'e equations P_I-P_V. We also introduce a new rigorous and constructive method of regularization, elimination of singularities whose position and type are known. If these are unknown, the same procedure enables a highly sensitive resonance method to determine the position and type of a singularity. In applications where less explicit information is available about the Riemann surface, our approach and techniques lead to new approximate, but still much more precise reconstruction methods than existing ones, especially in the vicinity of singularities, which are the points of greatest interest.
