Overview
This research investigates the presence of bump functions and polynomials within reproducing kernel Hilbert spaces (RKHSs) derived from translation-invariant positive-definite kernels. The study identifies the decay rate of the kernel's spectral density as the determinant for whether these specific function types are included in the Hilbert space.
Approach
The methodology employed in this study integrates principles from harmonic analysis, specifically utilizing the uncertainty principle. Additionally, the Beurling-Malliavin theorem serves as a foundational tool for describing the conditions under which bump functions and polynomials are contained within the RKHSs under consideration. The focus is on translation-invariant kernels, a specific class of positive-definite kernels.
Findings
- The inclusion of bump functions in a reproducing kernel Hilbert space of a translation-invariant positive-definite kernel is determined by the decay characteristics of its spectral density.
- The inclusion of polynomials in such a Hilbert space is similarly controlled by how fast the spectral density of the kernel decays.
- The uncertainty principle of harmonic analysis and the Beurling-Malliavin theorem provide the theoretical framework for establishing these descriptive relationships.