arXiv subjects

Karina Gonzalez

Publications and source records attributed to Karina Gonzalez.

3 recordsLinked to original sources

arXiv2026

Reproducing Kernel Hilbert Spaces and entropy Kolmogorov numbers on compact Lie Groups

On a compact Lie group $G$, we consider the reproducing kernel Hilbert space $\mathcal{H}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of $\mathcal{H}_K$ into the space $C(G)$ of continuous functions on $G$.

math.FA↗

arXiv2024

Entropy numbers of Reproducing Hilbert Space of zonal positive definite kernels on compact two-point homogeneous spaces

We present estimates for the covering numbers of the unit ball of Reproducing Kernel Hilbert Spaces (RKHSs) of functions on $M^d$ a d-dimensional compact two-point homogeneous space. The RKHS is generated by a continuous zonal/isotropic positive definite kernel. We employ the representation in terms of the Schoenberg/Fourier series expansion for continuous isotropic positive definite kernels, given in terms of a family of orthogonal polynomials on $M^d$. The bounds we present carry accurate information about the asymptotic constants depending on the dimension of the manifold and the decay or growth rate of the coefficients of the kernel. The results we present extend the estimates previously known for continuous isotropic positive definite kernels on the d-dimensional unit sphere. We present the weak asymptotic equivalence for the order of the growth of covering numbers associated to kernels on $M^d$ with a convergent geometric sequence of coefficients. We apply our estimates in order to present a bound for the covering numbers of the spherical Gaussian kernel, and to present bounds for formal examples on $M^d$.

math.FA↗

arXiv2021

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

In this paper, we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype to define a Weierstrass fractal kernel. We investigate the properties of the reproducing kernel Hilbert space (RKHS) associated with this kernel by presenting an explicit characterization of this space. In particular, we show that this space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.

math.FA↗