For the problem of broadband beamforming in a subregion of interest, a robust Frost beamforming algorithm is derived by reconstructing the signal subspace. The basic idea of the proposed algorithm is to extract the characteristic components of the signal of interest (SOI) from the estimated signal-plus-interference subspace by a matrix filter first, then employ these characteristic components to reconstruct the signal subspace, and finally construct a set of linearly constrained minimum variance (LCMV) constraints to protect the SOI components. Compared with some other robust Frost beamformers, the proposed algorithm has a significant advantage, i.e., its steering-angle and band are effective to match the SOI without prior information. Hence, the performance of the proposed algorithm is almost always close to the optimal value across the whole region of interest. Theoretical analysis and simulation results validate the effectiveness of the proposed algorithm.