Range of the Fourier transform on $L^1$

Question

It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^d)$ into, but not onto, $\overline{C_0^0}(\mathbb{R}^d)$, where the closure is taken in the $L^\infty$ norm. This is a consequence of the open mapping theorem, for instance.

My question is: what's an explicit example of a function in $\overline{C_0^0}(\mathbb{R}^d)$ which is not in the image of $L^1(\mathbb{R}^d)$ under the Fourier transform?

I would also like to know whether there is a useful characterization of $\mathcal{F}(L^1(\mathbb{R}^d))$.

Remark: it is easy to see that the Banach space $\overline{C_0^0}(\mathbb{R}^d)$ consists of all continuous functions $f$ on $\mathbb{R}^d$ such that $f(\xi)\rightarrow 0$ as $|\xi|\rightarrow\infty$.

Answer 1

For the first question I commented above that the function $\sqrt(1-|x|^2)$ extended to $0$ outside the unit ball is not the fourier transform of any integrable function in dimension 2 or higher. In dimension $1$ there's "Further results" of Chapter I in Introduction to Fourier Analysis of Stein. In case you don't have access to the book, this is the construction: Observe that $|\int_a^b \sin(x)/x\ dx| \leq C<\infty$ for any strictly positive $a$ and $b$. Now, if $f\in L^1$ and $F(f)$ is odd you have $F(f)(x) = \int f(t) \sin(xt)\ dt$ up to a multiplicative constant. Than it's easy to see from the previous estimate that $|\int_1^b F(f)(x)/xdx|\leq C'<\infty$ uniformly in $b$. So a function which is continuous, odd and which decays too slowly ($1/\log(x)$ will do) is not the Fourier transform of an integrable function.

Answer 2

It's not germane to your question, but I can't resist pointing out that it is very hard to exhibit any continuous linear bijection from $L^1$(sensible measure space) onto $C_0$(sensible topological space), and in fact if either space is infinite then I suspect this is never possible, just for reasons of Banach space geometry. Thus, although it doesn't help with what you want to look at, I thought it might be worth mentioning that one can know the answer to "is the FT onto?" must be "no", before looking for an example or using properties of the Fourier transform.

(My caveats are because I don't want to categorically state it can't be done, but in all cases I can think of no such bijection will exist. However, both my general measure theory and my general topology are not what they should be, so I can't remember how to do things precisely in the most general settings.)

Anyway. I claim that there is no continuous linear bijection between $L^1({\mathbb R}^d)$ and $C_0(X)$, where $X$ is locally compact Hausdorff (e.g. a metric space). The reason is that we have big powerful results telling us that

(i) every bounded linear operator from $C_0(X)$ to $L^1({\mathbb R}^d)$ is weakly compact;

(ii) if the identity map on a Banach space $E$ is weakly compact, then $E$ is reflexive;

(iii) $L^1({\mathbb R}^d)$ is not reflexive (ibid).

Unfortunately I can't locate a self-contained proof of the key fact (i). (It can be deduced as a corollary of a rather powerful, fundamental and beautiful result - due to some promising former student of Dieudonné and Schwartz, not sure if he ever went on to do anything important...)

Answer 3

See Page 68 (Theorem 3.2.2) in my notes on Fourier Analysis: https://home.iitm.ac.in/mtnair/FS-2018.pdf

Answer 4

Any characterization of $\mathcal{F}(L^1(\mathbb{R}^d))$, as one of my harmonic analysis professors put it, will get you immediate tenure in a math department that cares about harmonic analysis. It is a well known open question in the field.