On functional reproducing kernels

We show that even if a Hilbert space does not admit a reproducing kernel, there could still be a kernel function that realizes the Riesz representation map. Constructions in spaces that are the Fourier transform of weighted spaces are given. With a mild assumption on the weight function, we are able to reproduce Riesz representatives of all functionals through a limit procedure from computable integrals over compact sets, despite that the kernel is not necessarily in the underlying Hilbert space. Distributional kernels are also discussed.