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Abstract
This dissertation explores the discrete approximation of differential operators by Sinc methods. This leads into the areas of Sinc matrices and, more generally, Toeplitz matrices.
In the first part of the dissertation, we explore the Sine matrices [special characters omitted] used to approximate derivative operators and their properties as a subset of skew-symmetric Toeplitz matrices. We will prove a key invertibility result for these Sine matrices that covers all of the matrices, thereby definitively answering the open question of invertibility of the Sine matrices [special characters omitted] for odd values of n.
In the second part of the dissertation, we seek to solve the first order system of equations [special characters omitted] where f is a known vector-valued function in [special characters omitted]. To accomplish this, we apply Sinc methods to discretize the problem. This generates a Toeplitz system of equations to solve with the special Sine matrix I(1). In order to solve the system of equations given the unique properties of I(1) and matrices like it, we develop modified versions of three standard Toeplitz solvers that create a new class of hybrid routines set up to utilize the beautiful matrix/vector duality of the Toeplitz systems that these matrices produce.
The three categories of Toeplitz systems of equations that we address in this dissertation can be categorized as follows. (1) Yule-Walker equations Ty = −r: We solve TY = − R with a modified version of the Durbin method. This system of equations arises in the solution of the two remaining classes of problems. (2) General right hand side equations Tx = f: We solve these with a modified version of the Levinson method. (3) Inversion of T: We solve this with a modified version of the Trench method.
