A generalization of a result of G. Pólya and its application to a continuous extension of the de la Vallée Poussin means

Variation diminishing properties are established for the periodic kernels (1+cost)λ,λ=12+n,n∈Z+. On the real axis, there are related variation diminishing properties of the functions umsgnuumsgnu, which are the Green’s functions for the differential operator D(m+1)D(m+1).