Recall that $L^2(\mathbb R)$ decomposes into the direct sum of the eigenspaces of the Fourier transform corresponding to its four eigenvalues, namely the four fourth roots of unity. If $f\in L^2(\mathbb R)$ actually belongs to the Schwartz space of functions (namely $\x^mf^{(n)}(x)\_\infty$ is finite for all $m,n\in \mathbb N$), do the orthogonal projections of $f$ onto the four eigenspaces also lie in the Schwartz space?
Let F denote the Fourier transform, and let f be a given function. Then consider the decomposition $$f=(f_1+f_2+f_3+f_4)/4,$$ where $$f_1=f+Ff+F^2f+F^3f,$$ $$f_2=f+iFfF^2fiF^3f,$$ $$f_3=fFf+F^2fF^3f,$$ $$f_4=fiFfF^2f+iF^3f.$$ 

