Chapter 6 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. First, polynomial interpolation is carried out using both the monomial basis and the Lagrange-interpolation formalism, sped up via the barycentric formula. This includes a derivation of the error and an emphasis on using unequally spaced points (Chebyshev nodes). Second, cubic-spline interpolation is introduced. Third, a section is dedicated to trigonometric interpolation, carefully working through the conventions and formalism needed to implement one of the most successful algorithms ever, the fast Fourier transform (FFT). Fourth, the topic of linear least-squares fitting is tackled, including the general formalism of the normal equations. The second edition includes a substantive new section on statistical inference, covering both frequentist and Bayesian approaches to linear regression. Nonlinear least-squares fitting is covered next, including the Gauss-Newton method and artificial neural networks. The chapter is rounded out by a physics project, on the experimental verification of the Stefan-Boltzmann law, and a problem set. In addition to providing a historical background on black-body radiation, the physics project shows an example of nonlinear least-squares fitting.

This chapter begins with the study of trigonometric interpolation and the discrete Fourier transform. As a first application, the numerical integration of periodic functions is discussed. More detailed topics, like existence and uniqueness of trigonometric interpolants, as well as alias as convergence of trigonometric interpolation are discussed. An important practical tool, the fast Fourier transform (FFT) is then introduced. With this at hand the most rudimentary ideas of signal processing are presented.