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A new approach providing new theoretical results to the optimal far-field focusing of uniformly spaced arrays, subject to a completely arbitrary mask for sidelobe bounds, is presented and discussed. In both cases of linear or planar arrays (with rectangular boundaries), it is first shown that the problem can be formulated, without any loss to performances on the maximum, as a linear programming one, guaranteeing a globally optimal solution. Second, a sufficient uniqueness criterion for the solution of the overall problem is also developed, which shows how the solution may not be unique (as is actually the case) when planar arrays are considered. In addition, further globally effective optimization procedures are proposed for the latter case in order to optimize directivity, smoothness of excitations, or other performance parameters in the set of equivalent solutions. Last, an extension to planar arrays with a nonrectangular boundary is also given. A thorough numerical analysis confirms the effectiveness of the approach proposed and of the solution codes developed.