  Quantized tensor trains (QTTs) have recently emerged as a framework for the
numerical discretization of continuous functions, with the potential for
widespread applications in numerical analysis. However, the theory of QTT
approximation is not fully understood. In this work, we advance this theory
from the point of view of multiscale polynomial interpolation. This perspective
clarifies why QTT ranks decay with increasing depth, quantitatively controls
QTT rank in terms of smoothness of the target function, and explains why
certain functions with sharp features and poor quantitative smoothness can
still be well approximated by QTTs. The perspective also motivates new
practical and efficient algorithms for the construction of QTTs from function
evaluations on multiresolution grids.
