High accuracy trigonometric approximations of the real Bessel functions of the first kind

We construct high accuracy trigonometric interpolants from equidistant evaluations of the Bessel functions $${{J}_{n}}(x)$$ of the first kind and integer order. The trigonometric models are cosine or sine based depending on whether the Bessel function is even or odd. The main novelty lies in the fact that the frequencies in the trigonometric terms modelling $${{J}_{n}}(x)$$ are also computed from the data in a Prony-type approach. Hence the interpolation problem is a nonlinear problem. Some existing compact trigonometric models for the Bessel functions $${{J}_{n}}(x)$$ are hereby rediscovered and generalized.