In this paper, we study efficient beam coverage design for multi-antenna systems in both far-field and near-field cases. To reduce the computational complexity of existing sampling-based optimization methods, we propose a new low-complexity yet efficient beam coverage design. To this end, we first formulate a general beam coverage optimization problem to maximize the worst-case beamforming gain over a target region. For the far-field case, we show that the beam coverage design can be viewed as a spatial-frequency filtering problem, where angular coverage can be achieved by weight-shaping in the antenna domain via an inverse FT, yielding an infinite-length weighting sequence. Under the constraint of a finite number of antennas, a surrogate scheme is proposed by directly truncating this sequence, which inevitably introduces a roll-off effect at the angular boundaries, yielding degraded worst-case beamforming gain. To address this issue, we characterize the finite-antenna-induced roll-off effect, based on which a roll-off-aware design with a protective zoom is developed to ensure a flat beamforming-gain profile within the target angular region. Next, we extend the proposed method to the near-field case. Specifically, by applying a first-order Taylor approximation to the near-field channel steering vector (CSV), the two-dimensional (2D) beam coverage design (in both angle and inverse-range) can be transformed into a 2D inverse FT, leading to a low-complexity beamforming design. Furthermore, an inherent near-field range defocusing effect is observed, indicating that sufficiently wide angular coverage results in range-insensitive beam steering. Finally, numerical results demonstrate that the proposed FT-based approach achieves a comparable worst-case beamforming performance with that of conventional sampling-based optimization methods while significantly reducing the computational complexity.