If we treat fourier transform as an operator on $L^1(\mathbb{R})$, then its image under fourier transform is the set of continuous functions which will vanish at infinity.
It is well known that the fourier transform of signum function is
$$\mathcal{F} (sgn)(u) =\frac{2}{ui}. $$
I know that signum function is not integrable over the real line. So, to evaluate its fourier transform, one can use limiting argument, say a sequence of functions that converges to signum function, because fourier transform is a bounded linear operator, and hence is continuous.
What puzzles me is that when using continuity, don't we need to ensure that the fourier transform is defined on the limiting function?
In this case, fourier transform of signum function is not defined due to reason given above. If this is the case, how would one obtain formula of the fourier transform of signum function?
Best Answer
1. BOTTOM LINE UP FRONT: Treat it as a distribution.
Since the signum function is not integrable on $\mathbb{R}$, it may be useful to view it as a tempered distribution.
Such "generalized functions" are bounded linear functionals on a class of very-well-behaved functions called Schwartz functions. One of Laurent Schwartz's achievements was finding a collection $\mathcal{S}$ of functions on $\mathbb{R}^n$ such that the set of Fourier transforms of these functions is $\mathcal{S}$ itself. That put the original functions and their Fourier transforms on equal footing.
2. Fourier transform of a distribution
Why is this useful? It means that every tempered distribution has a Fourier transform that is also a tempered distribution. It also provides some useful notation to derive expressions and properties of the Fourier transform of a known tempered distribution.
Given any distribution $\mathsf{T}$ we write the result of applying it to a Schwartz function $\varphi$ as $\left<\mathsf{T},\varphi\right>$, but it is to be understood that this is not an inner product of two objects of the same kind. The Fourier transform of the distribution $\mathsf{T}$ is the distribution $\widehat{\mathsf{T}}$ for which \begin{equation} \left<\widehat{\mathsf{T}},\varphi\right> = \left<\mathsf{T},\widehat{\varphi}\right> \end{equation} for every $\varphi\in\mathcal{S}$, where $\widehat{\varphi}$ is the Fourier transform of $\varphi$. Since $\varphi\in\mathcal{S}$, $\widehat{\varphi}\in\mathcal{S}$, too.
3. Fourier transform of signum
How is this related to the signum function? If $\mathsf{T}$ is the signum function viewed as a distribution, then \begin{equation} \left<\mathsf{T},\varphi\right> = \int\textrm{sgn}(x)\varphi(x)dx. \end{equation} The Fourier transform of this distribution satisfies (or is defined by) \begin{equation} \begin{split} \left<\widehat{\mathsf{T}},\varphi\right> &=~ \left<\mathsf{T},\widehat{\varphi}\right>\\ &=~ \int\textrm{sgn}(x)\widehat{\varphi}(x)dx\\ &=~ -\int_{-\infty}^{0}\widehat{\varphi}(x)dx + \int_{0}^{\infty}\widehat{\varphi}(x)dx. \end{split} \end{equation}
4. Changing order of integration
Let's consider the integral for positive reals. The very good behavior of $\varphi$ allows changing orders of integration in many, many situations. \begin{equation} \begin{split} \int_{0}^{\infty}\widehat{\varphi}(x)dx &=~ \int_{0}^{\infty}\left[\int\varphi(k)e^{-ixk}dk\right]dx\\ &=~ \lim_{R\to\infty}\int_{0}^{R}\left[\int\varphi(k)e^{-ixk}dk\right]dx\\ &=~ \lim_{R\to\infty}\int\left[\int_{0}^{R}e^{-ixk}dx\right]\varphi(k)dk \end{split} \end{equation} We do something very similar for the negative reals. \begin{equation} \begin{split} -\int_{-\infty}^{0}\widehat{\varphi}(x)dx &=~ -\int_{-\infty}^{0}\left[\int\varphi(k)e^{-ixk}dk\right]dx\\ &=~\lim_{R\to\infty}-\int_{-R}^{0}\left[\int\varphi(k)e^{-ixk}dk\right]dx\\ &=~ \lim_{R\to\infty}-\int\left[\int_{-R}^{0}e^{-ixk}dx\right]\varphi(k)dk \end{split} \end{equation}
We now address the sum of the $R$-dependent integrals. \begin{equation} \begin{split} \int_{0}^{R}e^{-ikx}dx - \int_{-R}^{0}e^{-ikx}dx &=~ \left.\frac{e^{-ikx}}{-ik}\right|_{x=0}^{x=R} - \left.\frac{e^{-ikx}}{-ik}\right|_{x=-R}^{x=0}\\ &=~ \frac{1 - e^{-ikR}}{-ik} - \frac{e^{ikR} - 1}{-ik}\\ &=~ \frac{e^{ikR} + e^{-ikR}}{ik} - \frac{2}{ik} \end{split} \end{equation}
5. Singularity at $k = 0$; Riemann-Lebesgue Lemma
The $k$ in the denominator will be a problem at $k=0$. But we know that the original integrals coverge. We must consider the new one as the limit of integrals from $\epsilon$ to $\infty$ and from $-\infty$ to $-\epsilon$.
\begin{equation} \int_{|k|>\epsilon}\frac{e^{ikR} + e^{-ikR}}{ik}\varphi(k)dk = \int 1_{\{k:|k|>\epsilon\}}(k)\frac{\varphi(k)}{ik}\left(e^{ikR} + e^{-ikR}\right)dk \end{equation} For each $\epsilon >0$, the function $1_{\{k:|k|>\epsilon\}}(k)\frac{\varphi(k)}{ik}$ is in $L^1(\mathbb{R})$, so this integral is that function's Fourier transform evaluated at $\omega = R$ plus the same Fourier transform evaluated at $\omega = -R$. The Riemann-Lebesgue Lemma shows that if $f\in L^1(\mathbb{R})$, then $\lim_{|R|\to\infty}\widehat{f}(R) = 0$. Hence, these $R$-dependent terms vanish as $R\to\infty$.
It is worth noting that this shows that we must take the $R$-limit first and then take the $\epsilon$-limit. The opposite order would not work.
6. Cauchy Principal Value
We are left with \begin{equation} \lim_{\epsilon\to 0}2i\int_{|k|>\epsilon}\frac{\varphi(k)}{k}dk. \end{equation} This is the Cauchy Principal Value of this integral. This shows that we must interpret the Fourier transform of the signum function very carefully, but we can do it in the sene of distributions: if $\textrm{sgn}(x)$ is the signum of $x$, then \begin{equation} \widehat{\textrm{sgn}}(k) = 2i~\mathsf{PV}\left(\frac{1}{k}\right). \end{equation}