Chapter 6: In this chapter, we explore the role of orthonormal (orthogonal and normalized) vectors in an inner-product space. Matrix representations of linear transformations with respect to orthonormal bases are of particular importance. They are associated with the notion of an adjoint transformation. We give a brief introduction to Fourier series that highlights the orthogonality properties of sine and cosine functions. In the final section of the chapter, we discuss orthogonal polynomials and the remarkable numerical integration rules associated with them.

We study the problem of approximating the value of a (weighted) integral of a function. We introduce the concepts of a quadrature rule and its consistency. Interpolatory quadrature rules and their construction are then discussed. Their error analysis, based on notions of previous chapters, is then presented. Then an error analysis based on the Peano Kernel Theorem and scaling arguments is developed. Newton-Cotes formulas then are developed and analyzed. Then, composite quadrature rules are presented and their use is illustrated. Their analysis is presented then, on the basis of Euler-Maclaurin formulas. The chapter concludes with a discussion of Gaussian quadrature formulas, their properties and optimality.

A snapshot of the theory of orthogonal polynomials on the line or on the circle is presented from the point of view of the asymptotics of bounded point evaluation constants, touching on classical fields of approximation theory of real functions, as well as complex analytic functions of a single variable.