We present an innovative decomposition approach for computing the price and the hedging parameters of American knock-out options with a time-dependent rebate. Our approach is built upon: (i) the Fourier sine transform applied to the partial differential equation with a finite time-dependent spatial domain that governs the option price, and (ii) the decomposition technique that partitions the price of the option into that of the European counterpart and an early exercise premium. Our analytic representations can generalize a number of existing decomposition formulas for some European-style and American-style options. A complexity analysis of the method, together with numerical results, show that the proposed approach is significantly more efficient than the state-of-the-art adaptive finite difference methods, especially in dealing with spot prices near the barrier. Numerical results are also examined in order to provide new insight into the significant effects of the rebate on the option price, the hedging parameters, and the optimal exercise boundary.