Convergence problem of multiple Fourier series and Gauss circle problem

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K18731
• Research Category: Grant-in-Aid for Challenging Research (Exploratory)
• Allocation Type: Multi-year Fund
• Research Field: Analysis, Applied mathematics, and related fields
• Research Institution: Ibaraki University
• Principal Investigator: Nakai Eiichi 茨城大学, 理工学研究科(理学野), 教授 (60259900)
• Co-Investigator(Kenkyū-buntansha): 倉坪 茂彦 弘前大学, 理工学研究科, 客員研究員 (50003512) ; 藤間 昌一 茨城大学, 理工学研究科(理学野), 教授 (00209082)
• Project Period (FY): 2017-06-30 – 2022-03-31
• Keywords: フーリエ級数 / 調和解析学 / 解析的整数論

Outline of Final Research Achievements

The convergence problem of Fourier series has been largely solved by research up to the 1960s in the case of single-variable functions, but in the case of multivariable functions, there are still many things that are not yet understood. In recent years, in addition to the Gibbs phenomenon, the Pinsky phenomenon and the Kuratsubo phenomenon have been discovered, and the complexity of the multivariable Fourier series has become more apparent.
On the other hand, the Gauss circle problem is a problem to evaluate the error between the area of a circle and the number of lattice points in the circle. Gauss proved that the order of error is less than or equal to the power of 1/2 of the area of the circle. In 1915, Hardy conjectured that it would be as close as possible to the power of 1/4, but it remains unresolved today.
In this study, we have proven the equivalence of these two seemingly unrelated unresolved problems.

Free Research Field

基礎解析学

Academic Significance and Societal Importance of the Research Achievements

フーリエが熱伝導方程式を解いてから約200 年になる。ただし、フーリエの方法には不完全な部分があり、当時から問題点が指摘されていた。その問題点の中心的なもののひとつがフーリエ級数の収束問題である。一方、ガウスの円問題に関するHardy予想は100年来の未解決問題である。
本研究では、これら調和解析学の古典的問題と解析的整数論の難問という、一見無関係と思われる2つの未解決問題の密接な相互関係を明らかにした。このことは、単に大問題の解決に寄与するだけではなく、2つの分野相互に新しい研究手法をもたらす。