Hilbert transform and Fourier transform

Assume the following relationship between the Hilbert and Fourier transforms: $$\mathcal{H}(f) = {\mathcal{F}^{-1}}(-i ~ \text{sgn}(\cdot) \cdot \mathcal{F}(f)),$$

where $\displaystyle [\mathcal{H}(f)](x) \stackrel{\text{def}}{=} \text{p.v.} \frac{1}{\pi} \int_{- \infty}^{\infty} \frac{f(t)}{x - t} ~ d{x}$.

What happens when $f(x)$ is a distribution? We know that the Fourier transform exists for distributions, but what about the Hilbert transform?

For example, take $f(x) = x^{n}$. Its Fourier transform exists as the $n$-th derivative of the delta function $\delta(x)$. However, the integral $$\text{p.v.} \frac{1}{\pi} \int_{- \infty}^{\infty} \frac{x^{n}}{x - t} ~ d{x}$$ is divergent. :(

• You have taken the Fourier transform of $x^n$ as a distribution, but why do you treat the Hilbert transform of $x^n$ as a function[thus it is divergent]? – Yimin Feb 15 '13 at 17:39
• is $isgn$ actually $sign$? – rschwieb Feb 15 '13 at 17:45
• What is ‘$w$’, by the way? – Haskell Curry Feb 15 '13 at 17:46
• @rschwieb $i$ is correct: see, e.g., here. – user53153 Feb 15 '13 at 18:18
• @5pm Oh :) I thought it might be a simple transposition. Good thing I didn't change it! – rschwieb Feb 15 '13 at 19:08

The Hilbert transform is anti-self-adjoint. Therefore, it is natural to define it on distribution by passing $\mathcal H$ to the test functions, similar to "pass the hat" definition of the Fourier transform. In fact, the Wikipedia article already says this.
Since the stated relation between $\mathcal{F}$ and $\mathcal H$ holds for test functions, the duality-based definition implies that it holds for distributions as well.
The multiplication by "sign" (with or without a constant "$i$" in front) does not map Schwartz functions to Schwartz functions, so that the otherwise-appealing notion of defining an automorphism on (tempered?) distributions by defining it on (Schwartz?) test functions does not succeed here.
In the case at hand, multiplication of a distribution by "sign" would make sense, for example, if the singular support does not contain $0$. Thus, a tempered distribution whose Fourier transform has singular support not including $0$ has a Hilbert transform.