Project/Area Number  10640145 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  HOKKAIDO UNIVERSITY 
Principal Investigator 
INOUE Akihiko Hokkaido Univ., Grad. School of Sci., Asso. Pro., 大学院・理学研究科, 助教授 (50168431)

CoInvestigator(Kenkyūbuntansha) 
NAKATA Toshio Fukuoka Univ. of Education, Dept. of Information Education, 教育学部, 助手 (10304693)
MIKAMI Toshio Hokkaido Univ., Grad. School of Sci., Asso. Pro., 大学院・理学研究科, 助教授 (70229657)
ARAI ASAO Hokkaido Univ., Grad. School of Sci., Pro., 大学院・理学研究科, 教授 (80134807)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 1999 : ¥1,500,000 (Direct Cost : ¥1,500,000)
Fiscal Year 1998 : ¥1,800,000 (Direct Cost : ¥1,800,000)

Keywords  Tauberian theorem / Mercerian theorem / Fourier transform / partial autocorrelation function / stationary time series / Fourier series / Abelian theorem / Hankel transform / タウバー型定理 / マーサー型定理 / フーリエ変換 / 偏相関関数 / 定常時系列 / フーリエ級数 / アーベル型定理 / ハンケル変換 / 長時間記憶 
Research Abstract 
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 paivariation plays an important role there.
