Representation theorems in finite prediction and linear-time algorithms for Toeplitz systems

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03654
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Hiroshima University
• Principal Investigator: Inoue Akihiko 広島大学, 先進理工系科学研究科(理), 教授 (50168431)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 有限予測 / テプリッツ系 / テプリッツ行列の逆行列 / 明示公式 / Baxter 型定理

Outline of Final Research Achievements

We consider a block Toeplitz matrix that has a symbol with an integrable inverse. Such a Toeplitz matrix corresponds to a minimal multivariate stationary process. Inoue showed that the inverse of the Toeplitz matrix has a simple representation in terms of the finite prediction of the dual process. Furthermore, Inoue derived explicit formulas for the inverse of the Toeplitz matrix using the representation formula. Inoue demonstrated the importance of the explicit formulas by two applications. The first is a strong convergence result, called a Baxter-type theorem, for the Toeplitz system. The second application is a closed-form formula for the inverse of a Toeplitz matrix with a rational symbol. The closed-form formula implies a linear-time algorithm for solving the Toeplitz system.

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Academic Significance and Societal Importance of the Research Achievements

テプリッツ行列は数学に限らず様々な分野で重要な役割を果たしている。このようなテプリッツ行列の逆と双対過程あるいは有限予測とを初めて結び付けた。また、テプリッツ行列の逆に対し、有限予測に基づく新しい解析手法を導入した。Baxter 型定理という概念を導入し、短期記憶と長期記憶の両方の定常過程の場合の対応するテプリッツ系に対しその Baxter 型定理が成り立つことを示した。有理シンボルを持つテプリッツ系を線形時間で解く超高速アルゴリズムを示した。テプリッツ行列の逆に対する局所評価の観点を導入した。