Positive definiteness of real quadratic forms resulting from the variable-step approximation of convolution operators

TL;DR

This paper introduces a new algebraic framework to verify the positive definiteness of quadratic forms from variable-step convolution kernels, aiding stability analysis of non-uniform time-stepping schemes.

Contribution

It provides the first general algebraic conditions for positive definiteness of variable-step convolution kernels, simplifying stability analysis.

Findings

Established algebraic criteria for positive definiteness.

Proved the positive definiteness for relevant convolution kernels.

Enabled straightforward stability analysis for non-uniform schemes.

Abstract

The positive definiteness of real quadratic forms with convolution structures plays an important role in stability analysis for time-stepping schemes for nonlocal operators.In this work, we present a novel analysis tool to handle discrete convolution kernels resulting from variable-step approximations for convolution operators. More precisely, for a class of discrete convolution kernels relevant to variable-step time discretizations,we show that the associated quadratic form is positive definite under some easy-to-check algebraic conditions. Our proof is based on an elementary constructing strategy using the properties of discrete orthogonal convolution kernels and complementary convolution kernels. To the best of our knowledge, this is the first general result on simple algebraic conditions for the positive definiteness of variable-step discrete convolution kernels. Using the unified theory, the stability for some simple non-uniform time-stepping schemes can be obtained in a straightforward way.