A transference principle for sampling operators on finite intervals

The powerful connection between the classical sampling series and the exponential sampling series is well-established, linked by a logarithmic change of variable. This transference allows theoretical results from the Fourier domain to be mapped directly to the Mellin domain. In this paper, we first formalize this transference principle. We then posit that this principle can be generalized to other variable transformations. As a primary contribution, we apply this generalized principle using specific diffeomorphisms. We introduce the “trigonometric sampling series” for periodic-like intervals and the “algebraic sampling series” for finite intervals with boundary singularities. We provide explicit formulations and demonstrate that these operators inherit the convergence properties of the classical series. Numerical experiments confirm that the algebraic operators significantly outperform uniform sampling in reconstructing singular functions on bounded domains.