Polynomial Expansions

Abstract

Fourier series are not well suited to represent nonperiodic functions. Other basis functions are needed, and polynomials are a possible alternative since they are easy to handle computationally. However, it was demonstrated in Chap. 5 that standard polynomials with x ⁿ as basis functions are no good for higher degrees. Orthogonal polynomials are a better alternative, and they have played a central role in the development of classic applied mathematics and approximation theory. Later they became more obscure as a result of the introduction of piecewise polynomials that are more convenient for finite element methods. However, during the last decades there has been a remarkable renascence for orthogonal polynomials when it comes to numerical solution of PDE by spectral methods, as we shall see in Chap. 12. This is why we introduce them in this chapter.