Series of Fourier type with integer coefficients by systems of dilates and translates of one function in $L_p$, $p\geq 1$

We consider systems of functions obtained from dilates and translations of one function in the spaces $L_p (0,1)$, $1 \leq p < \infty$. We obtain results on the representation with respect to these systems with integer coefficients of the Fourier type series. The approximation of the elements of the spaces $L_p (0,1)$, $1 \leq p < \infty$, according to the proposed methods has the property of image compression, that is, many coefficients are zero in this case. These studies may be of interest to specialists in the transfer and processing of digital information.