Interpolation and Approximation
Dimitrios Mitsotakis · 2023
This chapter provides an introduction to interpolation and approximation. Interpolation is a method of constructing a function whose graph passes through a specified set of points. When the data are too noisy to interpolate with smooth functions, people proceed with approximating the dataset. Specifically, they generate a function whose graph is close to the data points without necessarily passing through them. Such a method is the least squares approximation. The reason for using polynomials to approximate functions is because of the Weierstrass Approximation Theorem . This theorem guarantees that every continuous function can be approximated arbitrarily well by a polynomial. The chapter discusses the method of least squares for approximating datasets. The representation of the interpolating polynomial using Lagrange polynomials is very convenient especially for theoretical considerations. On the other hand, the computation of Lagrange polynomials can be proved inefficient in practice. For this reason, the chapter presents an alternative representation of the same interpolating polynomial using Newton&s;s form.