On Chebyshev interpolation of analytic functions

This paper reviews the notion of interpolation of a smooth function by means of Chebyshev polynomials, and the well-known associated results of spectral accuracy when the function is analytic. The rate of decay of the error is proportional to ρ−N , where ρ is a bound on the elliptical radius of the ellipse in which the function has a holomorphic extension. An additional theorem is provided to cover the situation when only bounds on the derivatives of the function are known. 1 Review of Chebyshev interpolation The Chebyshev interpolant of a function f on [−1, 1] is a superposition of Chebyshev polynomials Tn(x), p(x) = N ∑ n=0 cnTn(x), which interpolates f in the sense that p(xj) = f(xj) on the Chebyshev grid xj = cos(jπ/N) for j = 0, . . . , N . The rationale for this choice of grid is that under the change of variable x = cos θ, the Chebyshev points become the equispaced samples θj = jπ/N . Unlike f , the function g(θ) = f(cos θ) is now 2π-periodic. Note that g(θ) inherits the smoothness of f(x). The samples g(θj) can be made to cover the whole interval [0, 2π] if we extend the range of j to be 0 ≤ j ≤ 2N − 1 (this corresponds to a mirror extension of the original samples.) The rationale for choosing Chebyshev polynomials is that Tn(cos θ) = cos(nθ), so that Chebyshev interpolation of f from f(xj), with 0 ≤ j ≤ N − 1, is nothing but interpolation by trigonometric polynomials of g from g(θj), with 0 ≤ j ≤ 2N − 1. This interpolant is built as follows. Start by submitting the 2N samples g(θj) to the discrete Fourier transform and back; this gives g(θj) = N−1 ∑