The function k : Ω×Ω→ R is called a kernel on Ω if the matrix (k(xi, xj))1≤i,j≤n is positive semidefinite for all positive integers n and all x1, . . . , xn ∈ Ω. It is well-known that if k is a kernel on Ω, then there exists a Hilbert space H̃ and Φ̃ : Ω→ H̃ such that k(x, x′) = 〈Φ̃(x), Φ̃(x)〉H̃. While H̃ and Φ̃ are not uniquely determined by k, the Hilbert space of functionsHk = {〈v, Φ̃(·)〉H̃ : v ∈ H̃} i...

Abstract: The aim of study of this article is to determine the solution of seventh order boundary value problem. The behavior of the induction motor is simulated by fifth order differential equation model and induction machine with two rotor circuits is represented by the seventh order differential equations. In this study, a Reproducing Kernel Method (RKM) for a class of seventh-order nonlinea...

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding the hypothesis error from the sum automatically. Following this direction, we ...

We study learning algorithms generated by regularization schemes in reproducing kernel Hilbert spaces associated with an -insensitive pinball loss. This loss function is motivated by the -insensitive loss for support vector regression and the pinball loss for quantile regression. Approximation analysis is conducted for these algorithms by means of a variance-expectation bound when a noise condi...

Throughout this paper by using the frame theory we give a short proof for atomic decomposition for weighted Bergman space. In fact we show that the weighted Bergman space L 2 a (dA α) admit an atomic decomposition i.e every analytic function in this space can be presented as a linear combination of " atoms " defined using the normalized reproducing kernel of this space .

We propose a novel variant of conjugate gradient based on the Reproducing Kernel Hilbert Space (RKHS) inner product. An analysis of the algorithm suggests it enjoys better performance properties than standard iterative methods when applied to learning kernel machines. Experimental results for both classification and regression bear out the theoretical implications. We further address the domina...

In this paper we extend the definition of generalized Sobolev space and subsequent theoretical results established recently for positive definite kernels and differential operators in the article [21]. In the present paper the semi-inner product of the generalized Sobolev space is set up by a vector distributional operator P consisting of finitely or countably many distributional operators Pn, ...

In this paper, we propose a relatively new semi-analytical technique to approximate the solution of nonlinear multi-order fractional differential equations (FDEs). We present some results concerning to the uniqueness of solution of nonlinear multiorder FDEs and discuss the existence of solution for nonlinear multi-order FDEs in reproducing kernel Hilbert space (RKHS). We further give an error a...