Wiener - Hopf Factorization and its Applications

College Of William And Mary, Williamsburg VA

Investigators

Abstract

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.