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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. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of some new non-linear subdivision and multiresolution schemes using the Lehmer means. Our main objective is to attain adaption close to discontinuities. We present theoretical, numerical results and applications for different schemes. The main theoretical result is related to the four-point interpolatory scheme, that we write as a perturbation of a linear scheme. Our aim is to establish a one-step contraction property that allows to prove the stability of the new scheme. Indeed with a one-step contraction property for the scheme of differences, it is possible to prove the stability of the 2D algorithm constructed using a tensor product approach. In this article, we also consider the associated three points cell-average scheme, that we will use to present some results for image compression, and a non-interpolatory scheme, that we will use to introduce an application to subdivision curves in 2D. These applications show that the use of the Lehmer mean is suitable for the design of subdivision schemes for the generation of curves and for image processing. [ABSTRACT FROM AUTHOR]