As a powerful tool, near-field acoustical holography (NAH) recognizes the sound source effectively. The traditional equivalent source method (ESM) calculated by the Tikhonov regularization method could be available in the low-frequency band. To improve the resolution quality of traditional ESM in the middle-frequency band and the high-frequency band, we introduced a fast iterative shrinkage/thresholding algorithm via Laplace norm for sound source identification based on the equivalent source method (LFISTA). In this paper, four methods including the Tikhonov regularization method, the monotonic two-step iterative shrinkage/thresholding (MTwIST), the fast iterative shrinkage/thresholding algorithm (FISTA), and the proposed method were compared for evaluation of performance. Both of the simulated and experimental results indicated that the proposed method identified the targeted sound sources in the entire frequency range more precisely than the Tikhonov regularization method did; the proposed method fixed the problem that MTwIST had the unqualified resolution and unstableness in the low-frequency band.