Representability in Momentum Space

Another argument for the sufficiency of combed generalized functions for the representation of physical quantities uses the representation of the physics in momentum space. This representation is mathematically equivalent to the use of the Fourier coefficients of the real generalized functions as a way to represent those functions. The representation of the physics in momentum space can always be constructed, and it is often found to have a more direct and profound physical meaning than the original representation in position space. This is so, in no small measure, because the momentum-space modes that arise in this way represent distributed quantities, and do not involve any attempt at localizing any objects with mathematically perfect precision. Examples of this are everywhere, ranging from normal modes of vibration in classical and quantum physics to the construction of states of particles in quantum field theory.

The functions giving the solutions of physical problems are usually solutions of boundary value problems of partial differential equations. One of the most common and standard ways to solve such problems is via the representation of the solutions in momentum space, which is equivalent to the representation of the functions involved by their sequences of Fourier coefficients. It follows at once that all solutions obtained in this way are combed generalized functions. As was shown in the appendices of [9], the mere application of the low-pass filters will usually improve rather than harm the representation of the physics by the mathematics, even if the $\epsilon \to 0$ limit is not actually taken at the end of the day. Moreover, when it is taken, since we are taking the $\epsilon \to 0$ limit of a filtered function, we always end up with a combed function.

Since two zero-measure equivalent generalized functions have exactly the same set of Fourier coefficients, and thus cannot be distinguished from each other by the use of those coefficients, it follows that if the physics can be represented in momentum space at all, then the two zero-measure equivalent generalized functions must represent exactly the same physics. If we add to this the fact that the Fourier series, if convergent at all, always converges to the combed function belonging to that class of zero-measure equivalent generalized functions, then we must conclude that the combed generalized functions are sufficient for the representation of the physics involved. The same conclusion can be drawn for every other way used to represent a generalized function by its sequence of Fourier coefficients, such as through the corresponding inner analytic function or through alternative expressions involving center series, which are just other trigonometric series. This is so because all such method of representation always produce combed functions.