Taylor级数的一致化与构造性解析延拓
Uniformization and Constructive Analytic Continuation of Taylor Series
摘要：
我们分析的问题，全球重建的功能，尽可能准确地，基于部分信息的形式截断幂级数在某些点上，和额外的解析属性。这种情况在应用中经常发生。最佳程序的问题是开放的，我们将其表述为一个适定的数学问题。它的解决方案产生了一种实用的方法，与现有技术相比，精确度得到了显着的提高。我们的程序是基于黎曼曲面的一致化。作为一个应用程序，我们表明，我们的程序可以实现广泛的一类非线性常微分方程的解决方案。我们找到了一种新的均匀化方法，用它来构造特殊函数所需的均匀化映射，包括Painlev'e方程P_I-P_V的解。我们还介绍了一种新的严格的构造性正则化方法，消除了已知位置和类型的奇点。如果这些都是未知的，相同的程序使一个高灵敏度的共振方法，以确定位置和类型的奇点。在应用程序中，不太明确的信息是关于黎曼曲面，我们的方法和技术导致新的近似，但仍然更精确的重建方法比现有的，特别是在附近的奇点，这是最感兴趣的点。
英文摘要：
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.
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