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The joint synthesis of the spatial power pattern and polarization of arbitrary arrays is addressed. Specifically, the proposed approach gives the solution to a frequently encountered problem, namely the array design (i.e., the determination of the radiating element weightings) to achieve a pattern that is arbitrarily upper bounded, while its polarization is optimized in a given angular region. Any state of polarization (elliptical, circular and linear) can be synthesized and there is no restriction regarding the array geometry and element patterns. The synthesis problem is rewritten as a convex optimization problem, that is efficiently solved using readily available software. This ensures the optimality of the proposed solution. Various numerical results are presented to validate the proposed method and illustrate its potentialities. The synthesis of a sequentially rotated array is first addressed. Then a linear array of equispaced randomly oriented dipoles is considered. Finally, a conformal and a planar array of patches, where the mutual coupling effects are considered, are synthesized to radiate a linear and a circular polarization.