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How does the finite fourier transform work exactly?

The transform of f(x) is

[itex]\widetilde{f}(\lambda_{n})[/itex] =[itex]\int[/itex][itex]^{L}_{0}[/itex] f(x) X[itex]_{n}[/itex] dx

If I had a 3d wave equation pde and I applied Finite fourier transform on the pde for

z(x,y,t)=X(x)Y(y)T(t)

f(x)=z[itex]_{xx}[/itex]

[itex]\tilde{z_{xx}(\lambda_{n})}[/itex] =[itex]\int[/itex][itex]^{L}_{0}[/itex] z[itex]_{xx}[/itex] X[itex]_{n}[/itex](x) dx

f(x)=z[itex]_{yy}[/itex]

[itex]\tilde{z_{yy}(\lambda_{n})}[/itex] =[itex]\int[/itex][itex]^{L}_{0}[/itex] z[itex]_{yy}[/itex] Y[itex]_{n}[/itex](y) dy

etc for T

or

is the transform the same for everything, namely:

f(x)=z[itex]_{xx}[/itex]

[itex]\tilde{z_{xx}(\lambda_{n})}[/itex] =[itex]\int[/itex][itex]^{L}_{0}[/itex] z[itex]_{xx}[/itex] X[itex]_{n}[/itex](x) Y[itex]_{n}[/itex](y) dx

or is it a double integral?

or do you apply finite fourier twice with respect to y and x.

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# Finite Fourier Transform on a 3d wave

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