SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 2040-2053, January 2021.
We present general criteria ensuring the positivity of quadratic forms of convolution type generated by sequences of real numbers. A sharp result is obtained in the case of completely monotone sequences. Applications to widely used approximations of fractional integral and differential operators, including convolution quadrature and L1 formula on uniform temporal meshes, are presented and new inequalities are established. The results are found to be fundamental in the investigation of the numerical stability for time-fractional phase-field models. It is shown through a standard energy stability analysis and without the use of a fractional Grönwall inequality that several numerical schemes satisfy discrete energy dissipation laws.

中文翻译：

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应用于相场方程的离散时间分数算子的正性

SIAM 数值分析杂志，第 59 卷，第 4 期，第 2040-2053 页，2021 年 1 月。
我们提出了确保由实数序列生成的二次形式的卷积类型的正性的一般标准。在完全单调序列的情况下获得了清晰的结果。介绍了广泛使用的分数积分和微分算子近似的应用，包括卷积求积和均匀时间网格上的 L1 公式，并建立了新的不等式。结果被认为是研究时间分数相场模型数值稳定性的基础。通过标准的能量稳定性分析表明，在不使用分数格伦沃尔不等式的情况下，几种数值方案满足离散能量耗散定律。