# What does the Fourier transform of an L-infinity function look like locally?

Question: What does an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ look like locally?

As formulated, the question might be a bit difficult to answer since the Fourier transform of a function fL(ℝ) is a distribution, and it is not easy to "write down" a distribution. So let me first illustrate the situation at hand with an easy example:

Example: The Fourier transform of the Heaviside function H(x) (i.e. the characteristic function of the positive reals) is given by a linear combination of the function 1/x and of the Dirac delta function (see this Wikipaedia entry for the exact formula, as well as for the meaning of the distribution "1/x").

The formalism of distributions is bit overkill for talking about measures, and things that look like 1/x. For example, the primitive of an element of $\mathcal F \big( L^\infty(\mathbb R)\big)$ is always a function (well defined outside of a set of measure zero). Using the above observation, we get the following

Reformulation of the question:
Let f(x) ∈ L(ℝ) be a function, and let g(x) be a primitive of its Fourier transform.
• What can g(x) look like locally?
• What local conditions must g satisfy in order to have a chance of coming from some fL(ℝ)?
• On what kind of sets can g fail to be continuous?

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It is pretty much the same as to describe the class $G$ of functions $g$ on the circle whose Fourier coefficients decay as $O(|k|^{-1})$. There is no nice "space side" property $P$ that would characterize them but for every nice "space side" property $P$ one can figure out in finite time if it holds for all such functions or not.
1) On the circle being in this class it is a local property (this needs compactness of the circle) because if $g\in G$, then the product of $g$ and any sufficiently smooth function is in $G$ and you can do partitions of unity.
2) There are obvious inclusions $BV\subset G\subset BMO$ ("bounded variation" and "bounded mean oscillation"). If you need something tighter than that, tell the family of comparison spaces you want to use.
3) Since $\sum_{k\ge 1} \frac 1kz^k$ is unbounded at $1$ and continuous everywhere else, we can move such spikes around to create a function that is locally unbounded on any closed set we want and discontinuous on any $F_\sigma$ set we want including the entire circle.