This paper describes a study of the physical limitations of adaptive antenna arrays using analytical methods. It is shown how the antenna pattern of an N-element array can be calculated from the complex array weights. The available degrees of freedom of the array can be used to null up to N-1 jammers while maintaining gain in the direction of the desired signal. Alternatively, some degrees of freedom can be used to specify other properties of the antenna pattern. As an example, the antenna pattern of a four element two-dimensional array with large, nonuniform element spacing is calculated for the case of three jammers and a desired signal and shown to be superior to the pattern obtained with uniform spacing. It is then shown how further improvements can be obtained by using some of the available degrees of freedom to constrain the first derivative of the pattern in the vicinity of the desired signal. This method produces a "maximally flat" antenna response and eliminates some of the adverse effects of grating lobes. Finally, some other applications of this methodology are mentioned.