The paper deals with stability and convergence of a spectral approximation scheme when applied to solving the unsteady incompressible two- or three-dimensional Navier-Stokes equations in the primitive variables. The solution is sought for a certain time interval under the assumption that all the given and the sought functions are \(2\pi\)- periodic in the first variable \(x_1\). Additionally, the non-slip boundary conditions for velocity are imposed at \(x_1= 1\) and \(x_1= -1\). Fourier spectral approximation in the periodic directions and Legendre spectral approximation in the non-periodic one are used. Heavy functional analytic apparatus prevails in the paper.