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.

翻译：我们基于谱碰撞极限和多项式射流几何，发展了阵列超方向性的再生核希尔伯特空间解释。当$M$元线性阵列的间距趋于零时，线性阵列生成的指数族发生谱碰撞，关联的有限维子空间在再生核意义下收敛到多项式射流空间。阵列增益等于再生核的对角线评估，而$M^2$端射定律源于Christoffel-Darboux核的端点渐近行为。与依赖于近奇异优化的经典推导不同，本方法将阵列增益极限与数值条件分离，并将超方向性识别为几何边界集中现象：硬边界处Christoffel函数的坍缩速度比内部快$M$倍。二次缩放特指平坦$L^2([-1,1])$几何结构；其他再生核希尔伯特空间几何允许不同的集中缩放。