Geodesic
Integer expansion in systems of~translates and dilates of~a single function
Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 796-806

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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.
Keywords: functional systems of translates and dilates of a single function in the multidimensional spaces $ L_p \{ (0,1]^m \}$, $1 \leq p < \infty$, multidimensional Fourier-type series, multidimensional Fourier-type series with integer coefficients, digital information processing, digital information transfer, integer expansions of functions. 
@article{IM2_2020_84_4_a7,
     author = {V. I. Filippov},
     title = {Integer expansion in systems of~translates and dilates of~a single function},
     journal = {Izvestiya. Mathematics },
     pages = {796--806},
     publisher = {mathdoc},
     volume = {84},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a7/}
}
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V. I. Filippov. Integer expansion in systems of~translates and dilates of~a single function. Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 796-806. http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a7/