Estimates for the quantized tensor train ranks for the power functions

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 $k=1,2,3,\dots ,2^d$. 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.