Tauberian and Mercerian theorems for Fourier transforms with applications

Inoue和Bingham发现，如果适当地推广到系统中，比例默塞利定理可以应用到陶伯利定理的证明中。这意味着最初在研究默塞耳定理中发展起来的技术也适用于研究陶伯利定理。利用这一思想，他们成功地证明了一般积分变换的德哈恩的陶伯利定理的类似物。他们也用同样的思想证明了解析数论中一些算术和的陶伯利定理。Inoue用Kasahara研究了平稳过程预测误差的渐近性。这个问题涉及到当数据数量增加时预测误差的渐近行为。利用这些结果，Inoue证明了偏自相关函数的渐近性公式。这个公式虽然早前就被推测出来了，但除了一些微不足道的情况外，甚至在特殊情况下也没有得到证明。关键思想是使用加权三角逼近的结果来证明必要的陶培尔条件。井上与菊池一起证明了傅立叶级数和积分的陶培尔问题的一个开放问题。这个问题是由于鲍亚士。这个结果也可以看作是1995年井上结果的自然延伸。证明的思想是使用归纳法将问题简化到可以使用Inoue和Bingham关于Hankel变换的早期结果的情况。配对变异的概念在这里起着重要的作用。

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.