To stay with finite (as opposed to infinite) degrees of freedom, Goldstein might have a complex discrete Fourier transform (DFT) $$ {\bf r}_j(t)~=~\sum_{k=1}^N e^{2\pi i jk/N }{\bf q}_j(t)$$ (or a real version thereof) in mind. Here the amplitudes ${\bf q}_j(t)$ play the role of generalized coordinates. This is for instance useful for finding normal modes of lattice models.

To stay within finite (as opposed to infinite) degrees of freedom (which is the topic of the book), Goldstein might have in mind a complex discrete (as opposed to continuous) Fourier transform $$ {\bf r}_j(t)~=~\sum_{k=1}^N e^{2\pi i jk/N }{\bf q}_j(t)$$ (or a real version thereof). Here the amplitudes ${\bf q}_j(t)$ play the role of generalized coordinates. This is for instance useful for finding normal modes in lattice models.