Asymptotic analysis on a non-standard Hilbert space of non-absolutely integrable functions

TL;DR

This paper investigates the KS-2 Hilbert space built via the Henstock-Kurzweil integral, analyzing its properties, integral operators, and asymptotic bounds for kernel embeddings, advancing understanding of non-absolutely integrable functions.

Contribution

It introduces the KS-2 space constructed with Henstock-Kurzweil integrals, proves a Mercer-type theorem, and derives asymptotic bounds for kernel embedding covering numbers.

Findings

Established a Mercer-type representation theorem for kernels in KS-2 space.

Derived asymptotic upper and lower bounds for covering numbers of RKHS embeddings.

Showed how Fourier coefficient decay influences embedding estimates.

Abstract

In this work, we study the Kuelbs-Steadman-2 space (KS-2 space), a Hilbert space constructed via the Henstock-Kurzweil integral, which allows handling non-absolutely integrable functions. We present the construction of the KS-2 space over measurable subsets of $\mathbb{R}^d$ and explore its functional properties with particular focus on integral operators associated with symmetric kernels. A Mercer-type representation theorem is established for such kernels in a KS-2 space, leading to the characterization of the associated Reproducing Kernel Hilbert Spaces (RKHS). As an application, we derive asymptotic upper and lower bounds for the covering numbers of the embedding of the RKHS into the KS-2 space, highlighting how the Fourier coefficients decay rate of the kernels influences the estimates.