Wiener - Hopf Factorization and its Applications

摘要：教授罗德曼和Spitkovsky将研究各种问题有关的矩阵和算子函数的维纳-霍普夫型及其应用的因式分解。其中包括：（1）概周期因子分解;（2）抽象交换群中的因子分解;（3）构造性Wiener-Hopf因子分解;（4）Corona型定理;（5）插值;（6）数值域;（7）非线性矩阵方程和离散问题;（8）逆波散射;（9）黎曼-希尔伯特问题;（10）金融数学。在AP分解问题中，PI将研究分解的存在性，并获得更完整和建设性（尽可能）的分解图，重点是具体的AP矩阵函数的一个和几个变量，特别是在应用中出现。众所周知的（在某些情况下）与电晕型定理的连接将进一步探讨，无论是在抽象阿贝尔群中的因子分解的抽象设置，相对于对偶群的总阶，还是在AP矩阵函数类的更具体设置中。相关的问题包括可逆性和Fredholness标准Toeplitz运营商与矩阵符号和有限部分的方法，这些运营商在适当的Besikovitch空间。PI还将关注Wiener-Hopf分解的几个应用领域。有限区间上的差分方程在逆散射中起着重要的作用，将通过对某个4 × 4矩阵函数族的因式分解来研究;在逆散射的另一方面，具有有限傅立叶谱的J-酉AP矩阵值多项式及其因式分解是重要的。Riemann-Hilbert问题和密切相关的主题（如正交函数）将研究非标准轮廓，包括自相交轮廓。这包括非标准轮廓上的黎曼-希尔伯特问题的解的参数依赖性。维纳-霍普夫分解也将用于设计二次收敛算法求解排队模型，特别是M/G/1马尔可夫链。理论上证明了算法的收敛性，数值实验验证了算法的有效性。 提出的研究成长的经典领域的分析和运营商理论。主题的选择既受应用的影响，又针对应用。 经典Wiener-Hopf分解在积分方程、偏微分方程和衍射理论中有着重要的应用。PI将继续研究其自然推广到几乎周期矩阵值函数（一个和几个变量），这是在考虑有限区间上的积分方程和逆散射和数学物理其他部分的相关问题时出现的。Riemann-Hilbertm问题及相关正交函数理论的预期结果将用于滤波器设计、图像压缩与分析以及多元随机过程。将研究新的应用，特别是金融数学，其中最近更充分的Levy过程模型的衍生品定价导致卷积方程的类型，维纳-霍普夫分解技术非常适合。预计将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。

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.