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This paper makes use of the Pisarenko framework, originally devised for temporal power spectrum estimation, to introduce a method for spatial power estimation that outperforms the beamforming method (except in extreme cases with serious calibration errors) as well as the Capon method (except in idealized situations with plentiful data and no miscalibration). An important feature of the proposed method is that it is user parameter-free, unlike most previous proposals with a similar character. Throughout the paper we emphasize a covariance matrix fitting approach to spatial power estimation, which provides clear intuitive explanations of the typical performance of the methods in the class under discussion. In a somewhat separated analysis, of interest for signal estimation applications, we derive the beamformer that passes a signal of interest in an undistorted manner, has minimum white-noise gain, and whose output power equals a given value (that should be larger than the Capon beamformer output power, which is known to have the smallest possible value). The given power value, referred to above, can be either obtained with a spatial power estimation method or perhaps provided directly by the user.