  We study the problem of representing a discrete tensor that comes from finite
uniform samplings of a multi-dimensional and multiband analog signal.
Particularly, we consider two typical cases in which the shape of the subbands
is cubic or parallelepipedic. For the cubic case, by examining the spectrum of
its corresponding time- and band-limited operators, we obtain a low-dimensional
optimal dictionary to represent the original tensor. We further prove that the
optimal dictionary can be approximated by the famous \ac{dpss} with certain
modulation, leading to an efficient constructing method. For the
parallelepipedic case, we show that there also exists a low-dimensional
dictionary to represent the original tensor. We present rigorous proof that the
numbers of atoms in both dictionaries are approximately equal to the dot of the
total number of samplings and the total volume of the subbands. Our derivations
are mainly focused on the \ac{2d} scenarios but can be naturally extended to
high dimensions.
