| Abstract: |
| The positive definiteness of discrete convolution kernels plays an important role in the stability analysis of time-stepping schemes for nonlocal models. Specifically, when these kernels are generated by convex sequences, their positivity can be verified by applying a classical result due to Zygmund. This talk has two main focuses. First, we improve Zygmund`s result and extend its validity to sequences that are almost convex. Next, we establish a more general inequality applicable to sequences that are nearly convex. Secondly, we consider convolution kernels on nonuniform grids and generalize the previous bounds. Our results are then applied to demonstrate the positivity properties of commonly used approximations for fractional integral and differential operators. |
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