On the completely positive kernels for nonuniform meshes

TL;DR

This paper investigates the complete positivity of convolutional kernels on nonuniform meshes, introducing a new pseudo-convolution operation and criteria, with applications to time fractional differential equations.

Contribution

It introduces the concept of pseudo-convolution for discrete kernels on nonuniform meshes and establishes criteria for their complete positivity.

Findings

Established a criterion for complete positivity of discrete kernels on nonuniform meshes

Introduced the pseudo-convolution operation for analyzing kernels

Applied the theory to L1 discretization of time fractional differential equations

Abstract

The complete positivity, i.e., positivity of the resolvent kernels, for convolutional kernels is an important property for the positivity property and asymptotic behaviors of Volterra equations. We inverstigate 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. Lastly, we apply our theory to the L1 discretization of time fractional differential equations on nonuniform meshes.