Chebyshev Polynomials as Basis Functions

Abstract

In the present chapter some of the important properties of Chebyshev polynomials are described, including their recursion relations, their analytic expressions in terms of the powers of the variable x, where $-1\le x\le 1$, and the mesh points required for the Gauss–Chebyshev integration expression described in Chap. 3. We also point out the advantage of the expansion into this set of functions, as their truncation error is spread uniformly across the $[-1,1]$ interval with the smallest error for functions that do not have strong singularities. The convergence of the expansion of functions in terms of Chebyshev polynomials will be illustrated, as well as the accuracy of the calculation of integrals and of derivatives. A novel “hybrid” method for calculating derivatives of higher order will also be described.