Convergence in Schwartz Space of Fourier transforms?

Question

Let $f_n$ a sequence in Schwartz space. Suppose that $\lim _{ n\rightarrow \infty }{ { \left\| { f }_{ n }-f \right\| }_{ \alpha ,0 }=0 } $, where ${ \left\| f \right\| }_{ \alpha ,0 }=\underset { x\in R }{ \sup } { \left| x \right| }^{ \alpha }\left| f\left( x \right) \right|$. Is it true that its Fourier transforms converge in the same sense, in others words $\lim _{ n\rightarrow \infty }{ { \left\| \widehat { { f }_{ n }-f } \right\| }_{ \alpha ,0 }= } 0$ or not?