Can anyone please help to clarify the sentences " into a fat tail part in L2 plus a fat body part in L1."

Question

In the link https://en.wikipedia.org/wiki/Fourier_transform#On_Lp_spaces what does this sentences mean: into a fat tail part in L2 plus a fat body part in L1? Would anyone please help?

Answer 1

The idea is that you first define the Fourier transform on $L^1$ and $L^2$. Then for $f\in L^p$ for some $p\in (1,2)$, write $$f(x)=\underbrace{1_{\left\{\left|f(x)\right|\geq 1\right\}}f(x)}_{f_1(x)}+\underbrace{1_{\left\{\left|f(x)\right|< 1\right\}}f(x)}_{f_2(x)}.$$

Then $f_1\in L^1$ is the tail part and $f_2\in L^2$ the body and you extend the Fourier transform to $L^p$ by linearity; $\hat{f}:=\hat{f}_1+\hat{f}_2$. It's an exercise to check that this definition is independent of the way in which you decompose $f$ into an $L^1$ and an $L^2$ part.