We propose a method for channel estimation in multiple-input multiple-output (MIMO) orthogonal frequency-division multiplexing (OFDM) wireless communication systems. The method exploits the band-sparsity of wireless channels in the delay-beamspace domain by solving a regularized optimization problem in a reproducing kernel Hilbert space (RKHS). A suitable representer theorem allows us to transform the infinite-dimensional optimization problem into a finite-dimensional one, which we then approximate with a low-dimensional surrogate. We solve the resulting optimization problem using a forward-backward splitting (FBS)-based algorithm. By exploiting the problem's modulation structure, we achieve a computational complexity per iteration that is quasi-linear in the number of unknown variables. We also propose a data-driven deep-unfolding based extension to improve the performance at a reduced number of iterations. We evaluate our channel estimators on ray-traced channels generated with SionnaRT. The results show that our methods significantly outperform linear methods such as linear minimum mean squared error (LMMSE) channel estimation based on aggregate channel statistics, both in terms of raw estimation accuracy as well as in downstream performance.