- #1
Gerenuk
- 1,034
- 5
I was trying to expand a three and more parameter functions similarly to the two-parameter case f(x,y)=(f(x,y)+f(y,x))/2+(f(x,y)-f(y,x))/2.
Anyway, to do the same for more parameters I need to solve
[tex]
\begin{pmatrix}
1 & 1 & 1 & \dotsb & 1\\
1 & \omega & \omega^2 & \dotsb & \omega^{n-1}\\
1 & \omega^2 & \omega^4 & \dotsb & \omega^{2(n-1)}\\
1 & \omega^3 & \omega^6 & \dotsb & \omega^{3(n-1)}\\
\vdots & &&& \vdots \\
1 & \omega^{n-1} & \omega^{2(n-1)} & \dotsb & \omega^{(n-1)(n-1)}
\end{pmatrix}\mathbf{x}=
\begin{pmatrix}
1 \\ 0 \\ 0 \\ 0 \\ \vdots \\ 0
\end{pmatrix}
[/tex]
with [itex]\omega=\exp(2\pi\mathrm{i}/n)[/itex]
Is there a closed form expression for x?
EDIT: Oh, silly me. I realized it's a discrete Fourier transform. So is this the correct way to expand then? In the 3 parameter case the solution would be
[tex]
f(x,y,z)=\frac13(f(x,y,z)+f(y,z,x)+f(z,x,y))+\frac13\left(f(x,y,z)+\omega f(y,z,x)+\omega^*f(z,x,y)\right)+\frac13\left(f(x,y,z)+\omega^*f(y,z,x)+\omega f(z,x,y)\right)[/tex]
with [itex]\omega=\exp(2\pi\mathrm{i}/3)[/itex]?
Now I'm just wondering why I get linear dependent terms when I consider the real part only?
Anyway, to do the same for more parameters I need to solve
[tex]
\begin{pmatrix}
1 & 1 & 1 & \dotsb & 1\\
1 & \omega & \omega^2 & \dotsb & \omega^{n-1}\\
1 & \omega^2 & \omega^4 & \dotsb & \omega^{2(n-1)}\\
1 & \omega^3 & \omega^6 & \dotsb & \omega^{3(n-1)}\\
\vdots & &&& \vdots \\
1 & \omega^{n-1} & \omega^{2(n-1)} & \dotsb & \omega^{(n-1)(n-1)}
\end{pmatrix}\mathbf{x}=
\begin{pmatrix}
1 \\ 0 \\ 0 \\ 0 \\ \vdots \\ 0
\end{pmatrix}
[/tex]
with [itex]\omega=\exp(2\pi\mathrm{i}/n)[/itex]
Is there a closed form expression for x?
EDIT: Oh, silly me. I realized it's a discrete Fourier transform. So is this the correct way to expand then? In the 3 parameter case the solution would be
[tex]
f(x,y,z)=\frac13(f(x,y,z)+f(y,z,x)+f(z,x,y))+\frac13\left(f(x,y,z)+\omega f(y,z,x)+\omega^*f(z,x,y)\right)+\frac13\left(f(x,y,z)+\omega^*f(y,z,x)+\omega f(z,x,y)\right)[/tex]
with [itex]\omega=\exp(2\pi\mathrm{i}/3)[/itex]?
Now I'm just wondering why I get linear dependent terms when I consider the real part only?
Last edited: