We consider multivariate L2-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error eL2−app,Λ(N, d) of all algorithms that use N information evaluations from the class Λ in the d-dimensional case. The two classes Λ considered in this paper are the class Λall consisting of all linear ...

Abstract We derive symmetric and antisymmetric kernels by symmetrizing antisymmetrizing conventional analyze their properties. In particular, we compute the feature space dimensions of resulting polynomial kernels, prove that reproducing kernel Hilbert spaces induced Gaussian are dense in functions, propose a Slater determinant representation kernel, which allows for an efficient evaluation eve...

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L := P∗T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional ad...

Let S be a polynomial of degree 2n + 2, that is, positive on the real axis, and let w = 1/S on (−∞,∞). We present an explicit formula for the nth orthogonal polynomial and related quantities for the weight w. This is an analogue for the real line of the classical Bernstein-Szegő formula for (−1, 1). 1. The result The Bernstein-Szegő formula provides an explicit formula for orthogonal polynomial...

We investigate the construction of all reproducing kernel Hilbert spaces of functions on a domain Ω ⊂ R that have a countable sampling set Λ ⊂ Ω. We also characterize all the reproducing kernel Hilbert spaces that have a prescribed sampling set. Similar problems are considered for reproducing kernel Banach spaces, but now with respect to Λ as a p-sampling set. Unlike the general p-frames, we pr...

This paper investigates the analytical approximate solutions of singular fourth order four-point boundary value problems using reproducing kernel method (RKM). The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve singular fourth order four-point boundary value p...

In a previous paper, we developed a general framework for establishing tractability and strong tractability for quasilinear multivariate problems in the worst case setting. One important example of such a problem is the solution of the Helmholtz equation −∆u + qu = f in the d-dimensional unit cube, in which u depends linearly on f , but nonlinearly on q. Here, both f and q are d-variate functio...

In this paper, an efficient Kernel based algorithm is developed with application in nonlinear system identification. Kernel adaptive filters are famous for their universal approximation property with Gaussian kernel, and online learning capabilities. The proposed adaptive step-size KLMS (ASS-KLMS) algorithm can exhibit universal approximation capability, irrespective of the choice of reproducin...