| Riemann zeta-values, Euler polynomials and the best constant of Sobolev inequality (2008) | |||||||||||||||||
Abstract | |||||||||||||||||
| MI: Global COE Program Education-and-Research Hub for Mathematics-for-Industry. グローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」. In this paper, we obtain the value $\sup_{u \in I_M , u \not \equiv 0} S_M (u)$, where $S_M (u)$ is the Sobolev functional and $I_M$ is a certain pre-Hilbert space. We also show that the functions attaining $\sup_{u \in I_M , u \not \equiv 0} S_M (u)$ are explicitly written by the Euler polynomials. This result is an analogue of the theorem proved by Kametaka, Yamagishi, Watanabe, Nagai and Takemura. Simultaneously, we obtain a much simpler proof of their theorem. | |||||||||||||||||
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