On functional reproducing kernels
Copyright policy )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 L 2 {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.
- Xuzhou University of Technology China (People's Republic of)
weighted l 2 spaces, Convolution, factorization for one variable harmonic analysis, weighted \(L^2\) spaces, positive definite, riesz representations, reproducing kernels, 47b34, Riesz representations, sobolev spaces, 42a85, Sobolev spaces, QA1-939, 46e35, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Kernel operators, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, 46e22, Mathematics
weighted l 2 spaces, Convolution, factorization for one variable harmonic analysis, weighted \(L^2\) spaces, positive definite, riesz representations, reproducing kernels, 47b34, Riesz representations, sobolev spaces, 42a85, Sobolev spaces, QA1-939, 46e35, Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces), Kernel operators, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, 46e22, Mathematics
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