Positivity of convolution quadratures generated by nonconvex sequences

The positive definiteness of real quadratic forms of convolution type plays an important role in the stability analysis of time-stepping schemes for nonlocal models. Specifically, when these quadratic forms are generated by convex sequences, their positivity can be verified by applying a classical result due to Zygmund. The primary focus of this work is twofold. We first improve Zygmund’s result and extend its validity to sequences that are almost convex. Secondly, we establish a more general inequality applicable to nonconvex sequences. Our results are then applied to demonstrate the positive definiteness of commonly used approximations for fractional integral and differential operators, including the convolution quadrature generated by the BDF2 formula. To conclude, we show that the stability of some simple time-fractional schemes can be obtained in a straightforward way.