In this work we provide theoretical estimates for the ranks of the power functions $f(k) = k^{-\alpha}$, $\alpha>1$ in the quantized tensor train (QTT) format for $k = 1, 2, 3, \ldots, 2^{d}$. Such functions and their several generalizations (e.~g. $f(k) = k^{-\alpha} \cdot e^{-\lambda k}, \lambda > 0$) play an important role in studies of the asymptotic solutions of the aggregation-fragmentation kinetic equations. In order to support the constructed theory we verify the values of QTT-ranks of these functions in practice with the use of the TTSVD procedure and show an agreement between the numerical and analytical results.

翻译：本文对幂函数 $f(k) = k^{-\alpha}$ ($\alpha>1$) 在量化张量列（QTT）格式中，当 $k = 1, 2, 3, \ldots, 2^{d}$ 时的秩给出了理论估计。此类函数及其若干推广形式（例如 $f(k) = k^{-\alpha} \cdot e^{-\lambda k}, \lambda > 0$）在聚集-碎裂动力学方程渐近解的研究中起着重要作用。为支撑所构建的理论，我们利用TTSVD程序验证了这些函数QTT秩的实际值，并展示了数值结果与解析结果之间的一致性。