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

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 $L^p(\mathbb{R})$ ( $1\le p\le \mathrm{\infty }$) to $H^1(\mathbb{R})$. As applications, four examples of partial Hausdorff integrals are also given.