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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: