We know that if we have a function $f(x)$, and we call $g(\omega)$ its Fourier transform, then the Fourier transform of $x f(x)$ is
$$\imath \frac{\mathrm{d} g(\omega))}{\mathrm{d}\omega} $$
and viceversa, ${\mathrm{d}f(x)}/{\mathrm{d}x}$ becomes $\imath \omega g(\omega)$
How can I transform an operator, for example
$$L(u):= u+x^2 u+\frac{\mathrm{d}u}{\mathrm{d}x}$$
so that it automatically transforms the variable $x$ in a derivative, and $x^n$ in a derivative of order $n$, and viceversa?
The specific problem I'm solving is the following:
ds[u_]:= A.{D[u,s],D[u,a]}.{1,0}
da[u_]:= A.{D[u,s],D[u,a]}.{0,1} (*where A is a constant matrix*)
G1[t_,x_,y_]:= G0[t,x,y] Integrate[K[tt,x,y],{tt,t,T}]
where G0[t,s,a]
is a function already defined and K[tt,x,y]
is a polinomial in the variable $x$ and $y$ (so its integral is still a polinomial in $x$ and $y$.
I know the inverse Fourier transform of G0
, that is F0[t,s,a]
. What I need is to do is to substitute all the the $x^n$ terms in the polinomial with the operator ds[u]
applied $n$ times and similarly, $y^m$ with the the operator da[u]
applied $m$ times.
thanks