Scattered Data Histopolation in Averaging Kernel Hilbert Spaces

TL;DR

This paper introduces averaging kernel Hilbert spaces for histopolation, providing theoretical foundations, construction principles, and numerical experiments demonstrating the method's convergence and practical effectiveness.

Contribution

It develops a new kernel-based framework for histopolation using averaging kernel Hilbert spaces, including construction, characterization, and analysis of approximation properties.

Findings

Characterization of averaging kernels via Fourier transform

Conditions for unisolvence in histopolation scenarios

Numerical experiments showing convergence and effectiveness

Abstract

Kernel-based methods offer a powerful and flexible mathematical framework for addressing histopolation problems. In histopolation, the available input data does not consist of pointwise function samples but of averages taken over intervals or higher-dimensional regions, and these mean values serve as a basis for reconstructing or approximating the target function. While classical interpolation requires continuity of the underlying function, histopolation can be performed in larger function spaces. In the framework of kernel methods, we will introduce and study the so-called averaging kernel Hilbert spaces (AKHS's) for this purpose. Within this setting, we develop systematic construction principles for averaging kernels and provide characterizations based on the Fourier-Plancherel transform. In addition, we analyze several representative histopolation scenarios in order to highlight properties of this approximation method, including conditions for unisolvence and possible error estimates. Finally, we present numerical experiments that shed some light on the convergence behavior of the presented approach and demonstrate its practical effectiveness.