Can anyone sort of give a proof for "if a function is concentrated in a cube then its Fourier transform is "mainly" concentrated in its dual cube"?
Also, i have seen similar arguments several times(like in the argument of wave packet decomposition). However, i never find formal arguments to prove relation of support of frequency function and support of its oscillatory integrals. I would really appreciate it if someone can suggest some reference about such relation.
Best Answer
This statement is wrong. Fourier transform of the characteristic function of the unit cube in dimention 1 equals $$\int_{-1}^1e^{-itx}dx=2(\sin t)/t.$$ Where is it "concentrated"?