A new family of means: the generalized weighted power means and applications

Abstract

Our aim is to define a new mean having specific properties which are crucial in several applications. As examples of this kind of applications, we can mention subdivision and multiresolution schemes based on nonuniform meshes and the numerical solution of conservation laws. Specifically, we pay attention to three relevant properties: to remain close to the minimum of the two values, to give a closer value to the weighted arithmetic mean of the two values when they are already similar, and to accomplish a Lipchitz condition. We define a family of means depending on a parameter k that matches these requirements. Using these means, it is possible to obtain more accurate theoretical constants in some stability theorems, and therefore they are better fitted to what happens in real examples. We carry out examples in the fields of subdivision schemes and hyperbolic conservation laws.

Keywords:

• Weighted nonlinear means
• nonuniform meshes
• nonlinear subdivision schemes
• stability theorems
• conservation laws

AMS subject classifications:

• 41A05
• 41A10
• 65D17

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first four authors have been supported through project 20928/PI/18 (Proyecto financiado por la Comunidad Autónoma de la Región de Murcia a través de la convocatoria de Ayudas a proyectos para el desarrollo de investigación científica y técnica por grupos competitivos, incluida en el Programa Regional de Fomento de la Investigación Científica y Técnica (Plan de Actuación 2018) de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia) and by the National Research Project PID2019-108336GB-I00. The fifth author has been supported through project CIAICO/2021/227 (Proyecto financiado por la Conselleria de Innovación, Universidades, Ciencia y Sociedad digital de la Generalitat Valenciana) and by grant PID2020-117211GB-I00 funded by MCIN/AEI/10.13039/501100011033.