Fourier transform of operators

**Mauritzvdworm** · February 7th 2011, 04:37 AM

Say we let the operator $D:=\frac{1}{i}\patial$
It is then possible to wire this operator in the form $D=F|D|$ like the polar decomposition of the bounded operators on some Hilbert space.
Let $A$ be the algebra of continuous functions on the circle and consider a function $f\in A$ with Fourier transform
$\hat{f}(k)=\int^{1}_{-1}f(x)e^{-ikx}dx$

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

I can show that
$\hat{D}\hat{f}(k)=k\hat{f}(k)$
Then defining $|\hat{D}|\hat{f}(k):=|k|\hat{f}(k)$
we can find $\hat{F}$ which we can just think of as a matrix with $\pm 1$ along the diagonal

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

Any help will be greatly appreciated

**Mauritzvdworm** · February 8th 2011, 02:34 AM

Since $\hat{F}$ acts on $\hat{f}(k)$ as follows
$\hat{F}\hat{f}(k)=\left\{\begin{array}{cc}\hat{f}( k) &,k\geq 0 \\ -\hat{f}(k) &,k<0 \end{array}\right.$

we can find the inverse Fourier transform
$<br /> Ff(x)=\sum_k \hat{F}\hat{f}(k)e^{ikx}<br /> =\sum_{k\geq0}\hat{f}(k)e^{ikx}-\sum_{k<0}\hat{f}(k)e^{ikx}<br />$
$=\sum_{k\geq0}\int^{1}_{-1}f(y)e^{-ik(y-x)}dy-\sum_{k<0}\int^{1}_{-1}f(y)e^{-ik(y-x)}dy$
$=\int^{1}_{-1}f(y)\left(1+\sum_{k>0}\left(e^{-ik(y-x)}-e^{ik(y-x)}\right)\right)dy$
$=\int^{1}_{-1}f(y)\left(1-2i\sum_{k>0}\sin k(y-x)\right)dy$

The derivative should come frome the commutator $[F,\pi(f)]\psi$ where $\pi: C(\mathbb{T})\rightarrow B(L^2(\mathbb{T}))$

where $\pi(f)\psi(x)=f(x)\psi(x)$

However, the details still elude me... my guess is that we should have a result that looks more or less like

$[F,\pi(f)]\psi \sim \frac{\psi(x)-\psi(y)}{x-y}$
together with some multiplicative factor.

Thread: Fourier transform of operators

LinkBack

Thread Tools

Display

Fourier transform of operators

Similar Math Help Forum Discussions

DE using fourier transform

Fourier transform

Laplace transform and Fourier transform what is the different?

MAPLE: graph in hz, fourier transform, wavelet transform

from fourier transform to fourier series

Search Tags