# Approximation of Hausdorff operators

## TL;DR

This paper studies how to approximate the adjoint of Hausdorff operators using Fourier transform truncation, providing explicit approximation rates and comparisons with identities for Lipschitz functions.

## Contribution

It derives formulas for approximation rates of Hausdorff operator adjoints in different metrics, including explicit rates for Lipschitz functions.

## Key findings

- Explicit formulas for approximation rates in various metrics.
- Comparison with approximate identities for Lipschitz functions.
- Analysis of the effect of truncation parameters on approximation quality.

## Abstract

Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate the adjoint to that Hausdorff operator of the given function. We find the formulas for the rate of approximation in various metrics in terms of the parameter of truncation and the components of the Hausdorff operator. Explicit rates of approximation and comparison with approximate identities are given in the case of Lipschitz $\alpha$ continuous functions.