Integer expansion of elements from spaces of nonintegrable functions into Fourier-type series in systems of contractions and shifts of one function

We present results on the expansion of elements of the spaces $L_p(0,1)$, $0 <p <1 $, into systems of functions that are contractions and shifts of one function. New algorithms are designed for the expansion of functions into Fourier-type series in these systems with integer coefficients. The expansions produced by the proposed methods have the property of image compression, i.e., a large number of expansion coefficients are zeros. The results may be of interest to specialists in the transmission and processing of digital information and to other researchers who have a need to expand nonintegrable functions into series in systems of contractions and shifts of one function.