Say we let the operator $\displaystyle D:=\frac{1}{i}\patial$

It is then possible to wire this operator in the form $\displaystyle D=F|D|$ like the polar decomposition of the bounded operators on some Hilbert space.

Let $\displaystyle A$ be the algebra of continuous functions on the circle and consider a function $\displaystyle f\in A$ with Fourier transform

$\displaystyle \hat{f}(k)=\int^{1}_{-1}f(x)e^{-ikx}dx$

Then we have

$\displaystyle \hat{Df}(k)=\hat{D}\hat{f}=\frac{1}{i}\int^{1}_{-1}\frac{\partial}{\partial x}f(x)e^{-ikx}dx$

I can show that

$\displaystyle \hat{D}\hat{f}(k)=k\hat{f}(k)$

Then defining $\displaystyle |\hat{D}|\hat{f}(k):=|k|\hat{f}(k)$

we can find $\displaystyle \hat{F}$ which we can just think of as a matrix with $\displaystyle \pm 1$ along the diagonal

I would like to find the inverse Fourier transform of $\displaystyle \hat{F}$, it should resemble something that looks like a derivative, but alas I cannot find it.

Any help will be greatly appreciated