다변수 체비เช프 다항식을 이용한 희소 보간법

📝 원문 정보

• Title: Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials
• ArXiv ID: 2001.09144
• 발행일: 2020-01-27
• 저자: Evelyne Hubert, Michael F. Singer

📝 초록 (Abstract)

이 논문은 Chebyshev 다항식을 일반화하고, 이 다항식들을 사용하여 특정 문제를 해결하는 방법에 대해 설명합니다. 특히, Chebyshev 다항식은 기존의 단변수에서 다변수로 확장되며, 이러한 다항식들이 어떻게 표현론과 Lie 대수와 연결되는지 살펴봅니다. 또한 Weyl 군이 Laurent 다항식 환에 작용하는 방식을 통해 Chebyshev 다항식들의 정의와 성질을 설명합니다.

💡 논문 핵심 해설 (Deep Analysis)

This paper discusses the generalization of Chebyshev polynomials and how they can be applied to solve various problems in a multivariate context. Specifically, Chebyshev polynomials are extended from univariate to multivariate settings using Lie algebra representation theory and Weyl groups acting on Laurent polynomial rings.

Summary: The paper generalizes Chebyshev polynomials to a multivariate setting and provides methods for their application in various contexts.

• Problem Statement: Existing Chebyshev polynomials are widely used but lack a systematic approach when extended to multiple variables. This paper aims to generalize these polynomials and provide methodologies applicable to multivariate settings.
• Solution (Core Technology): The paper defines Chebyshev polynomials using Lie algebra representation theory and Weyl groups acting on Laurent polynomial rings. It shows how these polynomials naturally arise in the context of Lie algebras, providing a foundation for their extension to multiple variables.
• Key Results: The paper systematically organizes the generalized form of Chebyshev polynomials and their properties, demonstrated through various examples. It provides clear insights into how these polynomials appear in Lie algebra representation theory.
• Significance and Applications: This work broadens the applicability of Chebyshev polynomials to multivariate settings, which is particularly useful in fields requiring a deep understanding of Lie algebras and representation theory.

📄 논문 본문 발췌 (Translation)

Reference