We give representations for solutions of time-fractional differential equations that involve operators on Lebesgue spaces of sequences defined by discrete convolutions involving kernels through the discrete Fourier transform. We consider finite difference operators of first and second orders, which are generators of uniformly continuous semigroups and cosine functions. We present the linear and algebraic structures (in particular, factorization properties) and their norms and spectra in the Lebesgue space of summable sequences. We identify fractional powers of these generators and apply to them the subordination principle. We also give some applications and consequences of our results.

我们给出了时间分数阶微分方程的解的表示形式，该时间分数微分方程的解涉及涉及离散核卷积的Lebesgue空间上的算子，这些卷积涉及通过离散傅里叶变换的核。我们考虑一阶和二阶有限差分算子，它们是一致连续半群和余弦函数的生成器。我们介绍了可加序列的Lebesgue空间中的线性和代数结构（特别是分解特性）及其范数和谱。我们确定这些生成器的分数功率，并将从属原理应用于它们。我们还给出了一些应用和结果的后果。