The Hirsch function and its properties

TL;DR

This paper introduces the Hirsch function, explores its properties, characterizes functions that are Hirsch functions, and provides a formula for reconstructing the original function from its Hirsch function.

Contribution

It defines the Hirsch function for continuous functions, analyzes its properties, and offers a method to recover the original function from its Hirsch function.

Findings

Hirsch function generalizes the h-index for continuous functions.

Characterization of functions that are Hirsch functions.

A formula for reconstructing the original function from its Hirsch function.

Abstract

The Hirsch function of a given continuous function is a new function depending on a parameter. It exists provided some assumptions are satisfied. If this parameter takes the value one, we obtain the well-known h-index. We prove some properties of the Hirsch function and characterize the shape of general functions that are Hirsch functions. We, moreover, present a formula that enables the calculation of f, given its Hirsch function $h_f$.