Density of sums of shifts of a single function

The report is devoted to the questions of approximation theory in functional spaces connected with the problem of existence of a function with dense sums of shifts: existence of a function whose sums of shifts are dense in Hardy spaces in the upper half-plane; density of derivatives of simple partial fractions with poles in the lower half-plane in Hardy spaces in the upper half-plane; existence of functions on $d$-dimensional Euclidean space, on $d$-dimensional torus and on $d$-dimensional integer lattice whose sums of shifts are dense in real spaces $L_p$ on these sets; complete description of compact Abelian groups on which there exists a function whose sums of shifts are dense in the $L_2$ norm in the corresponding real space of functions with zero mean.