Open Mathematics (Aug 2023)
On functional reproducing kernels
Abstract
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 L2{L}^{2} 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.
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