Positive definiteness of real quadratic forms resulting from the variable-step L1-type approximations of convolution operators

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 L1-type 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 by using the properties of discrete orthogonal convolution kernels and discrete 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.