Superdirectivity as Boundary Concentration under Spectral Collision

Abstract

Array superdirectivity is traditionally derived through singular optimization of densely spaced antenna arrays. In this paper, we show that the phenomenon admits a geometric interpretation as a concentration effect induced by spectral collision. As the spacing of an $M$-element linear array tends to zero, the exponential family generated by a linear array undergoes a spectral collision, and the associated finite-dimensional subspaces converge in reproducing kernel to a polynomial jet space. The maximum achievable array gain equals the diagonal evaluation of the reproducing kernel, and is therefore governed by the reciprocal Christoffel function. For the classical flat $L^2([-1,1])$ geometry, the Christoffel--Darboux kernel exhibits boundary concentration, yielding the quadratic $M^2$ superdirective law as a direct consequence of kernel asymptotics. This viewpoint separates intrinsic gain limits from numerical conditioning and identifies superdirectivity as a manifestation of a more general concentration mechanism. The framework further shows that the classical $M^2$ scaling is not universal: alternative spectral geometries produce different concentration laws through their associated Christoffel asymptotics. The results establish a direct connection between superdirectivity, reproducing kernels, orthogonal polynomials, and concentration phenomena arising from singular spectral limits.