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An old idea to obtain a good Fourier series approximation in such cases is to seek to represent a smooth, nonperiodic function with a Fourier series on a larger, regular domain. This is known as a Fourier extension/continuation. With this in mind, the purpose of this talk is to show that one can compute Fourier extensions that possess many of the beneficial properties of classical Fourier series, including rapid convergence and good resolution properties for oscillatory functions. Moreover, although intrinsically ill-conditioned, it transpires that this approach is also numerically stable in practice.
In many problems of interest, one faces the situation where the particular function is only prescribed on an equispaced grid. Platte, Trefethen & Kuijlaars have recently shown that any exponentially convergent method for recovering an analytic function from its values of an equispaced grid must also be exponentially ill-conditioned. In the final part of this talk I will describe the use of Fourier extensions for this problem, and discuss how they relate to this theoretical barrier.
This is joint work with Daan Hubrechs (KU Leuven) and Jesus Martin-Vaquero (Salamanca).
