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From the co-array perspective, sparse spatial sampling can significantly increase the degrees-of-freedom (DOFs), enabling us to perform underdetermined direction-of-arrival (DOA) estimation. By leveraging the increased DOFs from the sparse spatial sampling, we develop a new underdetermined DOA estimation method for wideband signals, named wideband sparse spectrum fitting (W-SpSF) estimator. In W-SpSF, we formulate a sparse reconstruction problem that includes a quadratic weighted covariance fitting term added to a sparsity-promoting regularizer. Meanwhile, the optimal regularization parameter of W-SpSF is studied to ensure robust sparse recovery. Numerical results enabled nested arrays demonstrate that the W-SpSF estimator outperforms the spatial smoothing based MUSIC algorithm and works well in nonuniform noise environment.