One of the important things I typically emphasize in discussing the use of “smoothed sums” is that whichever function is used for this purpose is of little importance, to the point that writing down an explicit smoothing function is a terrible faute de goût (think of a Coke with confit de canard). As it turns out, I’ve recently had two opportunities to do (or almost do) the opposite for good reasons. So here are some exceptions to the rule… (I’ll add a similar discussion to the corresponding tricki article).
(1) In one case, I and my collaborators J. Wu and Y-K. Lau needed to smooth the characteristic functions of a product of intervals in a high-dimensional rectangle:
$latex X=[a,b]^n\subset [0,\pi]^n,$
where a and b where fixed, but the dimension n was growing, and in fact was the main uniformity parameter. This is a slightly unorthodox type of problem, but as it turned out, we were spared trying to make this work by hand because E. Carneiro mentioned a result of the right type in a short talk at the IAS, due to Barton, Montgomery and Vaaler. They use multi-dimensional versions of the Beurling-Selberg approximating trigonometric polynomials (very nicely reviewed here by Vaaler) to get very clean and controllable upper and lower bounds for the type of functions of interest; this was already used by Y. Lamzouri in a study of the distribution of ζ(1+it).
(2) In another ongoing work with A. Nikeghbali, we required smooth, compactly supported, approximations of the characteristic function of a ball (in fixed dimension this time, say an interval in the real numbers), with as good decay as possible of the Fourier transform at infinity:
$latex \hat{f}(t)\ll 1/G(|t|)$
with G growing as fast as possible. Any fixed compactly supported test function gives this with
$latex G(t)=c(1+|t|)^A$
for any A>0 and a suitable c=c(A), but our result would gain maximal applicability by using a function with better decay. At first, I misremembered rather shamefully the Paley-Wiener theorem, and claimed in a draft that one could get exponential decay: for some f at least (non-zero), I said one could take
$latex G(t)=c\exp(\alpha t)$
with c and α positive. Of course, the Paley-Wiener theorem doesn’t state anything remotely comparable, and when I realized this, I also realized that I didn’t really know the answer to the implied question: given f smooth, positive, with compact support, how fast can its Fourier transform decay? The problem is of course that this type of functions are hard to write down explicitly and so one can’t just compute some Fourier transforms by hand to get an idea (and for our problem, it seems hard to use, say, a Gaussian instead of function with compact support).
After some interesting searching around, it seems the following is true: for every δ>0, one can find a function f as above with
$latex \hat{f}(t)\leq c\exp(-|t|^{1/(1+\delta)}),$
and on the other hand, this is false with δ=0 (i.e., the foolish claim of exponential decay was a serious mistake…) As far as the existence statement goes, I used a nice argument in Hörmander’s volume 1 of The analysis of linear partial differential operators (which is mysteriously unknown to Google books), specifically Theorem 1.3.5 there, where he constructs compactly supported smooth functions as uniform limits of iterated convolutions of functions with support in
$latex [0,a_i]$
where
$latex a=\sum_{i\geq 0}{a_i}<+\infty.$
The resulting limit has support being in the interval [0,a], and although Hörmander doesn’t state a bound for the Fourier transforms, he gives estimates for the derivatives. So, using the standard way of bounding Fourier transforms using k integration by parts, one finds after optimizing the number k for given x that the Fourier transform decay as stated provided one selects
$latex a_i=1/(i+1)^{1+\delta}.$
Altogether, this is a reasonably explicit and handy way of constructing test functions with controlled decay of the Fourier transform. And since there is still a fair amount of genericity, it is still reasonably tasteful…
For the limitation of decay, I found a paper (and related results) of Beurling and Malliavin, and it seems (I have to look at this more carefully before feeling confident…) that this is a consequence of “standard” properties of entire functions of finite type (i.e., bounded by Cexp(a|z|) for some a and C; the Paley-Wiener theorem does state that the Fourier transform of a compactly-supported function has this property).
I had never heard of all this before, but this seems to be an active area of analysis; Beurling and Malliavin elucidated quite precisely the permitted decay/growth condition of Fourier transforms of measures with compact support; their main application is to the completeness of systems of exponentials: given a discrete set Λ of real or complex numbers, for which values of r is it true that the exponential functions
$latex x\mapsto e^{i\lambda x},\quad\quad \lambda\in\Lambda$
span the space
$latex L^2([-r,r])$
(with respect to Lebesgue measure)?
[Also, as it turns out, for the immediate applications Ashkan and I have in mind, we can be perfectly happy with (arbitrarily fast, but fixed) polynomial decay…]