  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.
