A Physical Interpretation for Finite Tight Frames

Though finite tight frames arise in many applications, they have often proved difficult to understand and construct. We investigate the nonlinear problem of finding a tight frame for which the lengths of the frame elements have been prescribed in advance. Borrowing several ideas from Classical Mechanics, we show that this problem has a natural, intuitive interpretation. In particular, we show that such frames may be characterized as the minimizers of a potential energy function, and justify their interpretation as “maximally orthogonal” sequences. By exploiting this idea, we are able to show that such frames always exist, provided the requisite lengths satisfy a “fundamental inequality.” In so doing, we characterize those sequences of nonnegative numbers which arise as the lengths of a tight frame’s elements.