Originally Posted by **Neighbor**

Yes, this is the case. For simplicity (and in order to avoid the "definiteness issue") I assumed the sequence to be finite. In this case, its Fourier transform is bounded and hence well-defined. Moreover, if the norm in question were the L2 norm, the Parseval theorem guarantees that the L2 norm of the sequence is equal to that of its (discrete) Fourier coefficients. However, it is possible to figure out (even approximately) what is the L1 norm of the sequence given only the Fourier coefficients of the latter?