Journal of Approximation Theory ( IF 0.9 ) Pub Date : 2020-03-05 , DOI: 10.1016/j.jat.2020.105397 Petr Chunaev
In this paper we solve Padé (i.e. multiple) and Prony (i.e. simple exponential) interpolation problems for the generalized exponential sums with equal weights: and is a fixed analytic function under few natural assumptions. The interpolation of a function by is due to properly chosen and , which depend on , and . The sums are related to the -sums and amplitude and frequency sums (also known as generalized exponential sums), i.e. correspondingly to which generalize many classical approximants and whose properties are actively studied.
As for the Padé problem, we show that and have similar constructions and rates of interpolation, whereas calculating requires less arithmetic operations. Although the Padé problem for is known to have a doubled interpolation rate with respect to and thus to , it can be however unsolvable in quite simple and useful cases and this may entirely eliminate the advantage of . We show that, in contrast to , the Padé problem for always has a unique solution. What is even more important, we also obtain several efficient estimates for and , valuable by themselves, and use them in further evaluating interpolation quality and in numerical applications.
The above-mentioned Padé problem and estimates provide a basis for managing the more interesting Prony problem for exponential sums with equal weights , i.e. when . We show that it is uniquely solvable and surprisingly and can be efficiently estimated. This is in sharp contrast to the case of well-known exponential sums .
中文翻译:
权重相等的广义指数和的插值
在本文中,我们解决了权重相等的广义指数和的Padé(即多次)和Prony(即简单指数)插值问题:和 是在少数自然假设下的固定分析函数。函数的内插 通过 是由于正确选择 和 ,取决于 , 和 。总和 与有关 -和,幅度和频率和(也称为广义指数和),即对应于归纳了许多经典近似值,并对其性质进行了积极研究。
关于Padé问题,我们证明 和 具有相似的构造和内插率,而计算 需要较少的算术运算。尽管Padé问题 已知相对于 因此 ,但是在非常简单和有用的情况下可能无法解决,并且这可能会完全消除 。我们证明,与,Padé问题 总是有独特的解决方案。更重要的是,我们还获得了一些有效的估计 和 本身具有价值,并将其用于进一步评估插值质量和数值应用。
上述Padé问题和估计值为管理权重相等的指数和更有趣的Prony问题提供了基础 ,即何时 。我们证明它是唯一可解决的并且令人惊讶 和 可以有效地估算。这与众所周知的指数和的情况形成鲜明对比 。