Abstract

The Malmquist-Takenaka basis is a rational orthogonal basis constructed by mapping the Laurent basis from the unit circle to the real line by a Möbius transformation and multiplying by a weight to ensure orthogonality. Over the last century its properties have piqued the interest of various researchers including Boyd, Weideman, Christov, and Wiener. Despite this history, the approximation theory of this basis still defies straightforward description. For example, it was shown by Boyd and Weideman that for entire functions the convergence of approximation is superalgebraic, but that exponential convergence is only possible if the function is analytic at infinity (i.e. at the top of the Riemann sphere—- quite a strong condition). Nonetheless, convergence can be surprisingly quick, and the main body of this talk will be the result that wave packets clearly cannot have exponentially convergent approximations, but they /initially/ exhibit exponential convergence for large wave packet frequencies with exponential convergence rate proportional to said frequency. Hence, O(log(|eps|) omega) coefficients are required to resolve a wave packet to an error of O(eps). The proof is by the method of steepest descent in the complex plane. This is joint work with Arieh Iserles and Karen Luong (Cambridge).