The complete positivity, i.e., positivity of the resolvent kernels, for convolutional kernels is an important property for the positivity property and asymptotic behaviors of Volterra equations. We inverstigate the discrete analogue of the complete positivity properties, especially for convolutional kernels on nonuniform meshes. Through an operation which we call pseudo-convolution, we introduce the complete positivity property for discrete kernels on nonuniform meshes and establish the criterion for the complete positivity. Lastly, we apply our theory to the L1 discretization of time fractional differential equations on nonuniform meshes.

翻译：完全正性（即预解核的正性）对于卷积核而言，是Volterra方程正性保持性质和渐近行为的重要特性。本文研究完全正性性质的离散模拟，特别是非均匀网格上卷积核的对应性质。通过引入一种称为伪卷积的运算，我们定义了非均匀网格上离散核的完全正性，并建立了完全正性的判别准则。最后，我们将该理论应用于非均匀网格上时间分数阶微分方程的L1离散化。