Hilbert kernel

The form given by (3.8.11) is also convenient in that it indicates immediately that /h must vanish in the instantaneous elastic limit. The point is that y x) becomes a delta function in this limit so that /h vanishes, because of the antisymmetry of the Hilbert kernel. We remark that in fact it vanishes in both limits discussed after (3.7.1). Let... 

Rosipal R, Trejo LJ. Kernel partial least squares regression in reproducing Kernel Hilbert space. J Machine Learning Res 2001 2 97-123. 

Alternatively, considered as an operator over the whole Hilbert space K has a nontrivial kernel (see Eq. (2.4)) and, thus, has no inverse. 

For an orthonormal basis set, all of the matrix elements Sgr equal zero. The basis defines a subspace of the total Hilbert space and a projection operator can be introduced through the kernel... 

The operator <, > is called the transform operator or the inner product. K ix) is called the kernel of the transform and can be regarded as a coordinate in the infinite dimensional Hilbert space, within which an inner product is defined. The equation describing the kernel K is called the associated eigenproblem. 

Fitting forms obtained from other motivations such as standard cubic-spline methods. Morse-spline and rotated Morse-spline interpolation methods, reproducing kernel Hilbert space interpolation methods, distributed approximating functionals, and hybrid methods combining spline fits with simple empirical functions. ... 

DMBE) methods. Regarding the interpolation techniques, we focus on the global reproducing kernel Hilbert space method,... 

Rosipal R, Trejo LJ (2002) Kernel partial least squares regression in reproducing Kernel Hilbert Space. J Mach Learn Res 2(2) 97 123. doi 10.1162/15324430260185556 Baskin II, Tikhonova IG, Palyulin VA, Zefirov NS (2003) Selectivity fields comparative molecular field analysis (CoMFA) of the glycine/NMDA and AMPA receptors. J Med Chem 46(19) 4063 069... 

Another general method for constructing PESs from ab initio methods, based on the framework of the reproducing kernel Hilbert space , has recently been proposed. This interpolation method yields globally smooth representations with continuous energies and derivatives. The method has been illustrated for two- and three-atom systems, but can be extended to higher dimensions. 

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