Recent nonlinear "superresolution" techniques reported in the field of spectral analysis are of great interest in other fields as well, including radio-frequency (RF) adaptive array antenna systems. This paper is primarily a "cross-fertilization" treatise which takes the two most popular nonlinear techniques, the Burg maximum entropy method and the maximum likelihood method, and relates them to their similar nonlinear adaptive array antenna counterparts, which consist of the generic sidelobe canceller and directional gain constraint techniques. The comparison analysis permits an examination of their principles of operation from the antenna spatial pattern viewpoint, and helps to qualify, their actual superresolution performance. A summary of the resolution performance of several adaptive algorithms against multiple-incoherent sources is provided, including a universal graph of signal-to-noise ratio (SNR) versus source separation in beamwidths for the case of two equal-strength sources. Also, a significant dividend in the easy resolution of unequal-strength sources is reported. The superresolution of coherent spatial sources or radar targets is more difficult for these techniques, but successful results have been obtained whenever sufficient relative motion or "Doppler cycles" are available. Two alternate adaptive spatial spectrum estimators are suggested, consisting of a circular array predicting to its center point, and a new "thermal noise" algorithm.