Pointwise and uniform convergence. Examples from physics

I am a first-year mathematical student, and from a mathematical perspective I understand the difference between pointwise and uniform convergence of sequences and series of functions. However, I have been wondering about what phenomenons from physics (or other sciences using math) are described by sequences or series that converge pointwise, but not uniformly. Can you give any such examples? Thank you.

• For those of you unfamiliar with pointwise and uniform convergence, there's a very simple explanation here: youtube.com/watch?v=5rY0jCRW75Q. – SuperCiocia Aug 1 '15 at 13:54
• This question (v2) seems like a list question. – Qmechanic Aug 3 '15 at 9:18

Sure, there are a lot of them ! In my opinion, a basic conceptual reason for this is that most of the time, models used to describe a given class of physical phenomena have at their boundaries "uncontrollable" phenomena (otherwise we could extend the model straightforwardly and include them). So it is often the case that when you try to take some limits, the phenomena at these boundaries start to control the physics and you can't have uniform convergence, although you control the pointwise convergence in the bulk. The goal of physics is to understand how then to mix these new features and the old model into a new bigger framework.

Generally may observe pointwise convergence but not uniform where the Gibbs phenomenon occurs; in optics, acoustics, etc. In this case, I'm not sure that physics has much to say actually, this is just a property of Fourier transforms.

A very particular example in high energy physics is the well known pointlike limit of quantum string theory, where you recover the dynamics of quantum particles.

The quantum mechanical probability amplitude ("scattering amplitude") for a quantum string to scatter from a state to another tends towards the same amplitude for a particle to evolve in the same way when the length of the string goes to zero. The string amplitude is written as an integral over a certain finite dimensional space (the moduli space of Riemann surfaces with a given number of holes and marked points), and the essential contribution in this limit is localised at the boundary of this space. Pointwise you have a convergence to zero in the interior of the moduli space, but the integral itself has a nonzero limit, given by a boundary contribution. (to be precise, this statement is true only when the object in question -- this particle's scattering amplitude -- is "finite", see the problem of ultraviolet divergences and renormalization.).

Obviously this example is very specific, and the list is very very long.

Sciences use mathematics only as a tool. In almost all such applications, mathematical problems (such as pointwise vs uniform convergence) are not inherent to the scientific problem at hand, but arise from the mathematical model and are indicative of its limitations.

For example, when modelling a large collection of particles (be it in a solid (crystal) or fluid), one may describe their distribution via Dirac $\delta$-functions. Mathematically, this has certain issues (they are strictly not functions in the mathematical sense and not differentiable), which occur precisely because real particles are not point-like. But the point-like particle model may still be very useful in many applications.