Integer expansion in systems of translates and dilates of a single function

We study expansions with integer coefficients of elements in the multidimensional spaces $L_p\{(0,1]^m\}$, $1\leq p<\infty$, in systems of translates and dilates of a single function. We describe models useful in applications, including those in multimodular spaces. The proposed approximation of elements in $L_p\{(0,1]^m\}$, $1\leq p <\infty$, has the property of image compression, that is, there are many zero coefficients in this expansion. The study may also be of interest to specialists in the transmission and processing of digital information since we find a simple algorithm for approximating in $L_p\{(0,1]^m\}$, $1 \leq p < \infty$, having this property.