1999 Fiscal Year Final Research Report Summary
Tauberian and Mercerian theorems for Fourier transforms with applications
| Project/Area Number |
10640145
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Hokkaido University |
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Principal Investigator |
INOUE Akihiko Hokkaido Univ., Grad. School of Sci., Asso. Pro., 大学院・理学研究科, 助教授 (50168431)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Tauberian theorem / Mercerian theorem / Fourier transform / partial autocorrelation function / stationary time series / Fourier series / Abelian theorem / Hankel transform |
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| 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 pai-variation plays an important role there.
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