Analogues of the Hausdorff–Young and Hardy–Littlewood theorems

We study expansions of functions in the space $L^p$ with respect to systems similar to orthogonal ones. We find estimates for the coefficients and sufficient conditions on them under which the corresponding expansions converge in $L^p$. These results are analogues of the well-known Hausdorff–Young–Riesz and Hardy–Littlewood–Paley theorems in the theory of trigonometric and orthogonal series. It is shown that the resulting estimates are more exact than the classical ones even in the case of orthogonal systems.