Tauberian and Mercerian theorems for Fourier transforms with applications

傅立叶变换的陶伯里安定理和麦瑟里安定理及其应用

基本信息

批准号：
10640145
负责人：
INOUE Akihiko
金额：
$ 2.11万
依托单位：
Hokkaido University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640145/
关键词：
Tauberian theorem Mercerian theorem Fourier transform partial autocorrelation function stationary time series Fourier series Abelian theorem Hankel transform 長時間記憶

项目摘要

Inoue and Bingham found that ratio Mercerian theorems, if extended properly to systems, could be applied to the proofs of Tauberian theorems. This implies that the techniques developed originally in the study of Mercerian theorem are applicable to the study of Tauberian theorems. Using this idea, they succeeded in proving an analogue of de Haan's Tauberian theorem for general integral transforms. They also proved, using the same idea, Tauberian theorems for some arithmetic sums in analytic number theory.Inoue studied asymptotics for prediction errors of stationary processes with Kasahara. This problem concerns with the asymptotic behavior of the predition error when the number of data increases. Using the results, Inoue proved a formula on the asymptotics for the partial autocorrelation function. This formula, though conjectured earlier, had been unproven even for special cases, except for trivial ones. The key idea was to use a result on weighted trigonometric approximation to prove the necessary Tauberian condition.Inoue proved an open problem on Tauberian problems for Fourier seiries and integrals with Kikuchi. This problem was due to Boas. The result may also be seen as a natural extension to the result of Inoue in 1995. The idea of proof is to use an induction to reduce the problem to the case in which earlier results of Inoue and Bingham on Hankel transforms can be used. The notion of pai-variation plays an important role there.

Inoue 和 Bingham 发现，比率 Mercerian 定理如果适当地扩展到系统，可以应用于 Tauberian 定理的证明。这意味着最初在 Mercerian 定理研究中开发的技术也适用于 Tauberian 定理的研究。利用这个想法，他们成功地证明了一般积分变换的德哈恩陶伯定理的类似物。他们还用同样的思想证明了解析数论中一些算术和的陶伯定理。井上与笠原一起研究了平稳过程预测误差的渐近学。这个问题与数据数量增加时预测误差的渐近行为有关。利用这些结果，井上证明了偏自相关函数的渐近公式。这个公式虽然是早些时候推测出来的，但即使对于特殊情况也没有得到证明，除了一些微不足道的情况。其核心思想是利用加权三角近似的结果来证明陶伯条件的必要性。Inoue与Kikuchi一起证明了傅里叶级数和积分的陶伯问题的开放问题。这个问题是博阿斯造成的。该结果也可以被视为 1995 年 Inoue 结果的自然扩展。证明的想法是使用归纳法将问题简化为可以使用 Inoue 和 Bingham 关于 Hankel 变换的早期结果的情况。派变异的概念在那里发挥着重要作用。