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Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensiona ls=1/2 XY and XXZ models at T=∞ are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for the XXX model bear the unmistakable signature of spin diffusion in the form of a square root infrared divergence in the spectral density.