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The Nyquist-Shannon theorem states that the information accessible by discrete Fourier protocols saturates when the sampling rate reaches twice the bandwidth of the detected continuous time signal. This maximum rate (the NS-limit) plays a prominent role in Magnetic Resonance Imaging (MRI). Nevertheless, reconstruction methods other than Fourier analysis can extract useful information from data oversampled with respect to the NS-limit, given that relevant prior knowledge is available. Here we present PhasE-Constrained OverSampled MRI (PECOS), a method that exploits data oversampling in combination with prior knowledge of the physical interactions between electromagnetic fields and spins in MRI systems. In PECOS, highly oversampled-in-time k-space data are fed into a phase-constrained variant of Kaczmarz's algebraic reconstruction algorithm, where prior knowledge of the expected spin contributions to the signal is codified into an encoding matrix. PECOS can be used for scan acceleration in relevant scenarios by oversampling along frequency-encoded directions, which is innocuous in MRI systems under reasonable conditions. We find situations in which the reconstruction quality can be higher than with NS-limited acquisitions and traditional Fourier reconstruction. Besides, we compare the performance of a variety of encoding pulse sequences as well as image reconstruction protocols, and find that accelerated spiral trajectories in k-space combined with algebraic reconstruction techniques are particularly advantageous. The proposed sampling and reconstruction method is able to improve image quality for fully-sampled k-space trajectories, while allowing accelerated or undersampled acquisitions without regularization or signal extrapolation to unmeasured regions.