Approximation via partial Hausdorff integrals on $H^1(\mathbb{R})$

TL;DR

This paper extends the approximation of functions in the Hardy space using partial Hausdorff integrals by applying advanced multiplier and functional theories, providing new insights and examples in the context of harmonic analysis.

Contribution

It generalizes a known approximation result from Lebesgue spaces to the Hardy space using partial Hausdorff integrals, expanding the theoretical framework.

Findings

Extended approximation results from L^p spaces to H^1 space.

Provided four concrete examples of partial Hausdorff integrals.

Demonstrated the application of multiplier and K functional theories in this context.

Abstract

We obtain the result of approximating \( f \) in the \( H^1(\mathbb{R}) \) norm using partial Hausdorff integrals. Specifically, by leveraging the homogeneous multiplier theory of \( H^1(\mathbb{R}) \) and the \( K \) functional theory, one result from Pinos and Liflyand [CMB,~2021,~64,~no.3] is extended from \( L^p(\mathbb{R}) \) ( \( 1 \leq p \leq \infty \)) to \( H^1(\mathbb{R}) \). As applications, four examples of partial Hausdorff integrals are also given.