The Fluid Antenna System (FAS) overcomes the spatial degree-of-freedom limitations of conventional static antenna arrays in wireless communications.This capability critically depends on acquiring full Channel State Information across all accessible ports. Existing studies focus exclusively on narrowband FAS, performing channel estimation solely in the spatial domain. This work proposes a channel estimation and spatial equalization framework for wideband FAS, revealing for the first time an inherent group-sparse structure in aperture-limited FAS channels. First, we establish a group-sparse recovery framework for space-frequency characteristics in FAS, formally characterizing leakage-induced sparsity degradation from limited aperture and bandwidth as a structured group-sparsity problem. By deriving dictionary-adapted group restricted isometry property, we prove tight recovery bounds for a convex $\ell_1/\ell_2$-mixed norm optimization formulation that preserves leakage-aware sparsity patterns. Second, we develop a descending correlation group orthogonal matching pursuit algorithm that systematically relaxes leakage constraints to reduce subcoherence. This approach enables FSC recovery with accelerated convergence and superior performance compared to conventional compressive sensing methods like OMP or GOMP. Third, we formulate spatial equalization as a mixed-integer linear programming problem, complement this with a greedy algorithm maintaining near-optimal performance. Simulation results demonstrate the proposed channel estimation algorithm effectively resolves energy misallocation and enables recovery of weak details, achieving superior recovery accuracy and convergence rate. The SE framework suppresses deep fading phenomena and largely reduces time consumption overhead while maintaining equivalent link reliability.