Vallée Poussin kernels, shift-invariant subspaces and the spline connection

Shift-invariant approximation plays an important role in topics related with sampling theory. Despite being a theoretically mature topic, when it comes to practice, splines and Gaussians are the only well known generators. Here, we show that kernels introduced by Vallée Poussin, in context of the singular convolution integrals, can be successfully repurposed for shift-invariant approximation theory. Among other properties, such as excellent time-frequency localization, we prove that the Vallée Poussin kernel, or the VPK, admits partition of unity and forms a Riesz basis. Surprisingly, for a particular order, the VPK is identical to cubic splines at integer knots and this has interesting consequences. Quantitative Fourier analysis of the VPK, for convolution based approximation, shows a compelling performance and proximity to the spline family.