Let f be a complex-valued function of one real variable, continuous and compactly supported. Can it have a Fourier transform that is not Lebesgue integrable?



Does there exist a continuous function of compact support with Fourier transform outside L^1?
Here's a sketch of what I think is an example of the sort you want. Consider a trapezoidal function Tδ, supported on [-1,1], which is 1 on [-1+δ , 1-δ ] and is defined on the remaining intervals by interpolation in the obvious way. Then as δ tends to zero, the Fourier transform of Tδ is going to tend to infinity in the L1(R)-norm -- I can't remember the details of the proof, but since Tδ is a linear combination of Fourier transforms of Féjer kernels one can probably do a fairly direct computation.


Of course, the supremum norm of each Tδ is always 1. So the idea is to now stack scaled copies of these together, so as to obtain a function on [-1,1] which will be continuous (by uniform convergence) but whose Fourier transform is not integrable because its the limit of things with increasing L1-norm.


To be a little more precise: suppose that for each n we can find δ(n) such that Tδ(n) has a Fourier transform with L1-norm equal to n2 3n.


Put Sm = Σj=1m m-2 Tδ(m) and note that the sequence (Sm) converges uniformly to a continuous function S which is supported on [-1,1]. The Fourier transform of S certainly makes sense as an L2 function. On the other hand, the L1-norm of the Fourier transform of Sm is bounded below by


3m - (3m-1 + ... + 3 + 1) ~ 3m /2 


which suggests that the Fourier transform of S ought to have infinite L1-norm -- at the moment lack of sleep prevents me from remembering how to finish this off.


Alternatively, one could argue as follows. Consider the Banach space C of all continuous functions on [-1,1] which vanish at the endpoints, equipped with the supremum norm. If the Fourier transform mapped C into L1, then by an application of the closed graph theorem it would have to do so continuously, and hence boundedly. That means there would exists a constant M >0, such that the Fourier transform of every norm-one function in C has L1-norm at most M. But the functions Tδ show this is impossible.


