The method of regularization with the Gaussian reproducing kernel is popular in the machine learning literature and successful in many practical applications. In this paper we consider the periodic version of the Gaussian kernel regularization. We show in the white noise model setting, that in function spaces of very smooth functions, such as the infinite-order Sobolev space and the space of an...

A lot of problems the physical world can be modeled by non-linear ODE with their initial and boundary conditions. Especially higher order differential equations play a vital role in this process. The method for solution its effectiveness are as important modelling. In paper, on basis reproducing kernel theory, functions have been obtained solving some equations. Additionally, each problem homog...

We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the ...

Kernel functions are suitable tools for scattered data interpolation and approximation. We first review basic features of kernel-based multivariate interpolation, before we turn to the construction and the characterization of positive definite kernels and their associated reproducing kernel Hilbert spaces. The optimality of the resulting kernel-based interpolation scheme is shown. Moreover, we ...

Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between conditionally positive definite kernels and reproducing kernel Po...

The reproducing kernel function of a weighted Bergman space over domains in C is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kern...

We provide a new algorithm for a four-point nonlocal boundary value problem of nonlinear integro-differential equations of fractional order q ∈ 1, 2 based on reproducing kernel space method. According to our work, the analytical solution of the equations is represented in the reproducing kernel space which we construct and so the n-term approximation. At the same time, the n-term approximation ...