Convergence of a Fourier - Spline Representation for the Full - turn Map Generator

Single-turn data from a symplectic tracking code can be used to construct a canonical generator for a full-turn symplectic map. This construction has been carried out numerically in canonical polar coordinates, the generator being obtained as a Fourier series in angle coordinates with coe cients that are spline functions of action coordinates. Here we provide a mathematical basis for the procedure, nding su cient conditions for the existence of the generator and convergence of the Fourier-spline expansion. The analysis gives insight concerning analytic properties of the generator, showing that in general there are branch points as a function of angle and inverse square root singularities at the origin as a function of action. To be published in Proceedings of the Conference on Particle Beam Stability and Nonlinear Dynamics, Institute for Theoretical Physics, Santa Barbara, California, December 3-5, 1996, AIP Conference Proceedings Work supported in part by Department of Energy contract DE{AC03{76SF00515.