Wiener - Hopf Factorization and its Applications

维纳 - Hopf 分解及其应用

• 批准号: 0456625
• 负责人: Leiba Rodman
• 金额: --
• 依托单位: College of William and Mary
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456625&HistoricalAwards=false
• 关键词: Wiener; Hopf; Factorization; its; Applications

ABSTRACT: Professors Rodman and Spitkovsky will study a variety of problems concerning factorizations of matrix and operator functions of the Wiener-Hopf type and their applications. These include: (1) Almost periodic factorizations; (2) Factorization in abstract abelian groups; (3) Constructive Wiener-Hopf factorizations; (4) Corona type theorems; (5) Interpolation; (6) Numerical ranges; (7) Nonlinear matrix equations and queueing problems; (8) Inverse wave scattering; (9) Riemann-Hilbert problems; (10) Financial mathematics. In the AP factorization problems, the PIs will study existence of factorizations and obtain a more complete and constructive (whenever possible) factorization picture, with emphasis on concrete AP matrix functions of one and several variables arising in applications in particular. Well-known (in some cases) connections with corona type theorems will be further explored, both in the abstract setting of factorization in abstract abelian groups, with respect to a total order on the dual group, and in the more concrete setting of classes of AP matrix functions. Related issues include invertibility and Fredholmness criteria for Toeplitz operators with matrix symbols and finite section methods for these operators on suitable Besikovitch spaces. The PIs will also focus their attention upon several application areas of Wiener-Hopf factorization. Difference equations on a finite interval that play a role in inverse wave scattering will be studied via factorization of a certain family of four-by-four matrix functions; in another aspect of inverse scattering, J-unitary AP matrix valued polynomials with finite Fourier spectrum and their factorizations are of importance. Riemann-Hilbert problems and closely connected topics (such as orthogonal functions) will be studied with respect to non-standard contours, including contours with self-intersections. This includes parameter dependence of solutions of the Riemann-Hilbert problems on non-standard contours. Wiener-Hopf factorization will also be used to design quadratically convergent algorithms for solving queuing models, in particular M/G/1 Markov chains. The convergence will be proved theoretically, and effectiveness of the algorithms will be tested in numerical experiments. The proposed research grew out of classical areas of analysis and operator theory. The choice of topics is both influenced by and aimed to applications. Classical Wiener-Hopf factorization has been used as a powerful tool in integral equations, partial differential equations and diffraction theory. The PIs will continue their study of its natural generalization to almost periodic matrix valued functions (of one and several variables) which arises in consideration of integral equations on finite intervals and related problems in inverse scattering and other parts of mathematical physics. The expected results in the theory of Riemann-Hilbertmproblem and related orthogonal functions will be used in filter design, compression and analysis of images, and multivariate stochastic processes. Novel applications will be studied, in particular, financial mathematics, where the recent more adequate Levy processes models of derivatives pricing lead to convolution equations of the type that Wiener-Hopf factorization techniques are well-suited for. Interactions with scientists and engineers in diverse fields are anticipated. In addition, the PIs will also involve undergraduate students in their research.

摘要：Rodman 和 Spitkovsky 教授将研究有关 Wiener-Hopf 型矩阵和算子函数分解及其应用的各种问题。这些包括： (1) 几乎周期性因式分解； (2) 抽象阿贝尔群的因式分解； (3) 构造性 Wiener-Hopf 分解； (4) Corona型定理； (5)插值法； (6) 数值范围； (7) 非线性矩阵方程和排队问题； (8)逆波散射； (9)黎曼-希尔伯特问题； (10)金融数学。在 AP 因式分解问题中，PI 将研究因式分解的存在性，并获得更完整且更具建设性（只要可能）的因式分解图，重点是特别是在应用中出现的一个或多个变量的具体 AP 矩阵函数。将进一步探讨与 Corona 型定理的众所周知的（在某些情况下）联系，无论是在抽象阿贝尔群中因式分解的抽象设置中，相对于对偶群的全序，还是在更具体的类设置中AP 矩阵函数。相关问题包括具有矩阵符号的托普利茨算子的可逆性和 Fredholmness 准则以及这些算子在合适的贝西科维奇空间上的有限截面方法。 PI 还将重点关注维纳-霍普夫分解的几个应用领域。通过对某族 4×4 矩阵函数进行因式分解，研究在逆波散射中起作用的有限区间上的差分方程；在反散射的另一个方面，具有有限傅立叶谱的J-酉AP矩阵值多项式及其因式分解非常重要。将针对非标准轮廓（包括具有自交的轮廓）来研究黎曼-希尔伯特问题和密切相关的主题（例如正交函数）。这包括黎曼-希尔伯特问题的解对非标准轮廓的参数依赖性。 Wiener-Hopf 分解还将用于设计用于求解排队模型的二次收敛算法，特别是 M/G/1 马尔可夫链。从理论上证明算法的收敛性，并通过数值实验检验算法的有效性。 拟议的研究源于分析和算子理论的经典领域。主题的选择既受应用的影响，又是针对应用的。 经典维纳-霍普夫分解已被用作积分方程、偏微分方程和衍射理论中的强大工具。 PI 将继续研究其对几乎周期性矩阵值函数（一个或多个变量）的自然推广，该函数是在考虑有限区间积分方程以及逆散射和数学物理其他部分中的相关问题时产生的。黎曼-希尔伯特问题理论和相关正交函数的预期结果将用于滤波器设计、图像压缩和分析以及多元随机过程。我们将研究新的应用，特别是金融数学，其中最近更充分的衍生品定价 Levy 过程模型产生了维纳-霍普夫分解技术非常适合的类型的卷积方程。预计与不同领域的科学家和工程师进行互动。此外，PI 还将让本科生参与他们的研究。