Entropic inversion of Fourier transforms with incomplete data

Here we provide an entropic method to invert the Fourier transform of a bounded function defined on an interval, when the data consists of its sine and cosine transforms, but not necessarily of consecutive frequencies. The classical direct approach consists in just forming the linear combination of the trigonometric functions of the given frequencies multiplied by their respective coefficients. The problem is that this approach does not yield any information about the projection of the function on the space spanned by the missing frequencies. Our approach consists of regarding the Fourier inversion as an ill-posed linear inverse problem with box constraints, consisting of finding a function given a few of its sine and cosine transforms. To solve this problem, we propose a non-linear approach, consisting of minimizing an entropy function subject to the Fourier data as constraints. This approach provides us with a solution that has a non-vanishing projection on the space spanned by the Fourier coefficients in the original data set, from which a better approximation to the unknown function can be recovered. In addition to obtaining an explicit representation of the solution, we prove that the solution converges to the unknown function as the number of data points increases. Even though the reconstruction procedure is non-linear in the data, there is some quasi-linearity in the procedure.