Superdirectivity as a Spectral-Collision RKHS Limit

2026-06-06

AI Summary

This work elucidates the mathematical essence of the superdirectivity gain limit in array antennas by interpreting superdirectivity as a boundary concentration phenomenon in a reproducing kernel Hilbert space (RKHS). It analyzes the behavior of linear arrays as element spacing approaches zero through the lens of spectral collision limits and polynomial jet geometry. Leveraging the endpoint asymptotics of the Christoffel–Darboux kernel, the study demonstrates that the M² endfire gain scaling arises from the Christoffel function’s collapse rate at hard boundaries, which is M times faster than in the interior under the flat L²([-1,1]) geometry. This framework decouples the fundamental gain limit from numerical ill-conditioning and reveals that distinct RKHS geometries can yield different boundary concentration scalings.

Abstract

We develop a reproducing-kernel Hilbert space interpretation of array superdirectivity based on spectral-collision limits and polynomial jet geometry. 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. Array gain equals the diagonal evaluation of the reproducing kernel, and the $M^2$ endfire law emerges from endpoint asymptotics of the Christoffel-Darboux kernel. Unlike classical derivations that rely on near-singular optimization, the present approach separates array gain limits from numerical conditioning, and identifies superdirectivity as a geometric boundary concentration phenomenon: Christoffel function collapse at the hard edge is a factor of $M$ faster than in the interior. The quadratic scaling is tied specifically to the flat $L^2([-1,1])$ geometry; alternative RKHS geometries admit different concentration scalings.

Problem

• superdirectivity
• spectral collision
• reproducing-kernel Hilbert space
• array gain
• Christoffel function

Innovation

• superdirectivity
• reproducing-kernel Hilbert space
• spectral collision
• polynomial jet
• Christoffel-Darboux kernel