The Kalman filter is the Bayesian optimum solution to the problem of sequentially estimating the states of a dynamical system in which the state evolution and measurement processes are both linear and Gaussian. Given the ubiquity of such systems, the Kalman filter finds use in a variety of applications, e.g., target tracking, guidance and navigation, and communications systems. The purpose of this book is to present a brief introduction to Kalman filtering. The theoretical framework of the Kalman filter is first presented, followed by examples showing its use in practical applications. Extensions of the method to nonlinear problems and distributed applications are discussed. A software implementation of the algorithm in the MATLAB programming language is provided, as well as MATLAB code for several example applications discussed in the manuscript.

Gaussian quadrature is a powerful technique for numerical integration that falls under the broad category of spectral methods. The purpose of this work is to provide an introduction to the theory and practice of Gaussian quadrature. We study the approximation theory of trigonometric and orthogonal polynomials and related functions and examine the analytical framework of Gaussian quadrature. We discuss Gaussian quadrature for bandlimited functions, a topic inspired by some recent developments in the analysis of prolate spheroidal wave functions. Algorithms for the computation of the quadrature nodes and weights are described. Several applications of Gaussian quadrature are given, ranging from the evaluation of special functions to pseudospectral methods for solving differential equations. Software realization of select algorithms is provided. Table of Contents: Introduction / Approximating with Polynomials and Related Functions / Gaussian Quadrature / Applications / Links to Mathematical Software