Symmetrization inequalities on one-dimensional integer lattice

In this paper, we develop a theory of symmetrization on the one dimensional integer lattice. More precisely, we associate a radially decreasing function $u^*$ with a function $u$ defined on the integers and prove the corresponding Polya-Szeg\"{o} inequality. Along the way we also prove the weighted Polya-Szeg\"{o} inequality for the decreasing rearrangement on the half-line, i.e., non-negative integers. As a consequence, we prove the discrete weighted Hardy's inequality with the weight $n^\alpha$ for $1 < \alpha \leq 2$.