Let u be a function in the Schwartz class. Show that (F2u)(x)=(F(Fu))(x) =
2⇡u(x). Conclude that u is an even function if and only if F2u = 2⇡u. Then show that in
general (F4u)(x)=4⇡2u(x).

Let u be a function in the Schwartz class. Show that (F2u)(x)=(F(Fu))(x) =
2⇡u(x). Conclude that u is an even function if and only if F2u = 2⇡u. Then show that in
general (F4u)(x)=4⇡2u(x).