Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator P consisting of finitely or countably many distributional operators P n , which are defined on the dual space of the Schwartz space. The types of operators  consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators.  deduce that a certain appropriate full-space Green function G with respect to L := P *T P now becomes a conditionally positive function. In order to support this claim  ensure that the distributional adjoint operator P * of P is ll-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function G can be embedded into or even be equivalent to a generalized Sobolev space. As an application,  take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant s f,X to data values sampled from an unknown generalized Sobolev function f at data sites located in some set $${X \subset \mathbb{R}^d}$$.  provide several examples, such as Matern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of ll-known reproducing kernels are equivalent to a generalized Sobolev space. These examples further illustrate how  can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the “best” kernel function for kernel-based approximation methods.
['Gregory E. Fasshauer', 'Qi Ye']
I am writing a research paper on a idea Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators
[{"from": "human", "value": "I am writing a research paper on a idea Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator P consisting of finitely or countably many distributional operators P n , which are defined on the dual space of the Schwartz space. The types of operators  consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators.  deduce that a certain appropriate full-space Green function G with respect to L :=\u00a0P *T P now becomes a conditionally positive function. In order to support this claim  ensure that the distributional adjoint operator P * of P is ll-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function G can be embedded into or even be equivalent to a generalized Sobolev space. As an application,  take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant s f,X to data values sampled from an unknown generalized Sobolev function f at data sites located in some set $${X \\subset \\mathbb{R}^d}$$.  provide several examples, such as Matern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of ll-known reproducing kernels are equivalent to a generalized Sobolev space. These examples further illustrate how  can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the \u201cbest\u201d kernel function for kernel-based approximation methods."}]