The function $f(x)=1/(1+x^2)$ belongs to this class but not to the Schwarz Schwartz space of test functions.
Any $C^\infty$ function with bounded support belongs to the Schwarz Schwartz space but not to this class.
The function $f(x)=1/(1+x^2)$ belongs to this class but not to the Schwarz space.
Any $C^\infty$ function with bounded support belongs to the Schwarz space but not to this class.