On the completely positive kernels for nonuniform meshes

Abstract

The complete positivity for convolutional kernels is an important property for the positivity property and asymptotic behaviors of Volterra equations. We investigate the discrete analogue of the complete positivity properties, especially for convolutional kernels on nonuniform meshes. Through an operation which we call pseudo-convolution, we introduce the complete positivity property for discrete kernels on nonuniform meshes and establish the criterion for the complete positivity. We then apply our theory to the L1 discretization of time fractional differential equations on nonuniform meshes.