In which sense(s) do square integrable functions go to zero at infinity?

[gentzen]

In which sense(s) do square integrable functions go to zero at infinity?
Of course, they cannot go to zero at infinity in the sense of point evaluation, because point evaluation is not the appropriate concept for square integrable functions. There was a recent discussion in the Quantum Physics Forum, which focused on 1D functions:

No. This is non-relativistic, so there is no upper limit to the momentum. But the wave function must be square integrable (makes no difference if it is in position or momentum space), which requires that it goes to zero at infinity.

Square integrable is not enough to imply that it goes to zero at infinity.

Consider the function $f(x)$ defined as $f(x)=1$ if $x$ is integer, and $f(x)=0$ otherwise. It does not vanish at infinity but its integral is zero. Would you call such function artificial?

Consider a function that looks like little bumps above the x axis with height one. And the width of the bumps get smaller and smaller. Say 1/2, 1/4, 1/8, ... It will be integrable and it will not go to zero at infinity. Would you call this one artificial?

Your example is artificial, because point evaluation is not the important criteria in practice. And your example just artificially tries to hide this point, which was already exposed by Demystifiers example why such counterexamples are artificial.

If you look at the convolution of some arbitrary test function with your example function, the result goes to zero at infinity.

That your example was essentially no better than the example Demystifier already gave. If the problem is point evaluation, all another example can do is expose whether a certain technique to resolve that problem will work or not. The problem in Demystifier's example can be fixed by modifying the function on a set of measure zero. This no longer works for your example. For your example, for any $\epsilon >0$, one can modify the function on a set of measure $<\epsilon$. Fine, in this sense, your example is perhaps better. But this stuff with "modify functions on sets of measure ..." is still a bit mathematical and theoretical.

For square integrable functions, looking at results for test functions might be closer to what is relevant in practice. That is why I suggested that as criterion. Not sure whether it is a good one.

If it were just 1D functions, I am pretty sure that I could workout myself that both proposed criteria (a: the convolution with another square integrable test function goes to zero at infinity, b: you can modify the function on a set of arbitrarily small measure to get a function that goes to zero at infinity) work fine. But I vaguely remember that such questions are studied in the theory of Sobolev spaces, and that such function spaces become less well behaved in higher dimensions, so that often the $k$ in $W^{k,p}(\Omega)$ has to increase slightly as the dimension of $\Omega$ increases, to keep "the same good properties" near the boundary of $\Omega$.

(I also remember that math.stackexchange was very good at precisely answering such specific questions, together with extensive references to the literature. But I certainly don't want to post some discussion like the one above into a question at math.stackexchange.)

 

[martinbn]

You don't need to use point values. You can use the usual definition. A function vanishes at infinity if for every $\varepsilon>0$ there is a compact such that on the compliment the norm of the function is less than the $\varepsilon$. This would invalidate my example. But the original question was spesifically about functions and there values.

 

[gentzen]

You don't need to use point values. You can use the usual definition. A function vanishes at infinity if for every $\varepsilon>0$ there is a compact such that on the compliment the norm of the function is less than the $\varepsilon$.

This definition of "vanish at infinity" seems to work for square integrable functions in arbitrary dimension. Very good, so there is at least one well defined sense in which "square integrable functions go to zero at infinity".

This would invalidate my example. But the original question was spesifically about functions and there values.

And my question here is about arbitrary square integrable functions in arbitrary dimension. If I had further quibbles about the discussions in that other thread, I would have posted them there.

(This thread serves the purpose to put the question here in the context in which it arose, and also to hint at the meta question about how to ask questions that arose here at other sites, in case they cannot satisfactorily be answered here.)