  Based on an initial expectation from laboratory measurements or instrument
simulations, photometric passbands are usually subject to refinements. These
refinements use photometric observations of astronomical sources with known
spectral energy distribution. This work investigates the methods for and
limitations in determining passbands from photometric observations. A simple
general formalism for passband determinations from photometric measurements is
derived. The results are applied to the passbands of HIPPARCOS, Tycho, and Gaia
DR1. The problem of passband determination is formulated in a basic functional
analytic framework. For the solution of the resulting equations, functional
principal component analysis is applied. We find that, given a set of
calibration sources, the passband can be described with respect to the set of
calibration sources as the sum of two functions, one which is uniquely
determined by the set of calibration sources, and one which is entirely
unconstrained. The constrained components for the HIPPARCOS, Tycho, and Gaia
DR1 passbands are determined, and the unconstrained components are estimated.
