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.
From: Hong Yang [view email]
[v1]
Sat, 6 Jun 2026 13:39:48 UTC (327 KB)
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