How to generalize a theorem to 3D

[balugaa]

Homework Statement

So have this theorem defined for transforms on the Fourier kind, and hilbert spaces, on R2
The theorem sets bounds for norms of the functions and the transforms in the hilbert scale

I have the proofs for the 2D case given in my lecture notes

Problem: Generalize the above 2D theorem to prove a 3D result.

what does generalize the theorem mean?

Homework Equations

||f||_H(R2) <= ||RF||_H(Z) - here Z = cylinder

The Attempt at a Solution

So i have to use the generalisation to show

||f||_H(R3) <= ||Rf||_H(Z) - here Z = sphere

Does generalize mean i basically take the proof on the theorem above, and plug in the 3D case in the proof to derive the equations?

Basically this is what i would have done, just wanted to check that that's what generalize a theorem mean

 

[lanedance]

generalise often means to extend or make the theorem applicable to less restrictve conditions, in this case showing the R^2 result can be extended to R3

 

[balugaa]

generalise often means to extend or make the theorem applicable to less restrictve conditions, in this case showing the R^2 result can be extended to R3

Cool, i went all the way to R^n with the orig theorem, worked through the proof plugged in n=2 to show the 2D result .. and n = 3 for the 3D ...