We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system $\{u_n(x)\}^\infty_{n=0}$ generated by continuous periodic function $u(x)$. The result is based on Nikishin's example concerning convergence in measure. We also show that affine systems $\{u_n(x)\}^\infty_{n=0}$ generated by any Lipchitz function $u(x)$ are Besselian.