Basic Fourier Series: Convergence on and Outside the q-Linear Grid

A q-type Holder condition on a function f is given in order to establish (uniform) convergence of the corresponding basic Fourier series S-q[f] to the function itself, on the set of points of the q-linear grid. Furthermore, by adding other conditions, one guarantees the (uniform) convergence of S-q[f] to f on and "outside" the set points of the q-linear grid.