A new interpolation method for metric spaces based on bi-infinite sequences: The $R$-Method

TL;DR

The paper introduces the $R$-method, a novel metric interpolation technique based on bi-infinite sequences, which extends classical interpolation theory to nonlinear and metric settings, preserving key properties like Lipschitz continuity and compactness.

Contribution

It presents the $R$-method, a new intrinsic metric-based interpolation framework that generalizes classical methods and applies to nonlinear operators, preserving important analytical properties.

Findings

The $R$-method generates a new intrinsic metric on the intersection of spaces.

It embeds continuously into the $K_M$-interpolated space, linking to existing theory.

In normed settings, it acts as a genuine interpolation functor preserving Lipschitz properties.

Abstract

We introduce a new interpolation method for metric spaces, termed the $R$-method, based on bi-infinite linking sequences. Although the construction is inspired by the classical metric functional $J_M$, the resulting interpolated space is generated by a distinct object that behaves as a multiscale energy functional. This functional measures the minimal discrete action required to connect two points through $\mathbb{Z}$-indexed sequences, leading to a new intrinsic metric on $X_0 \cap X_1$. The associated interpolated space is obtained as the relative completion of this metric inside $X_0 \cup X_1$ and is genuinely different from those produced by the $J_M$- and $K_M$-methods. A fundamental structural property of the $R$-method is that the resulting space embeds continuously into the corresponding $K_M$-interpolated space, situating the construction naturally within the existing theory of metric interpolation. When the method is restricted to a normed setting, the $R$-method induces a genuine interpolation functor. In this framework, it preserves the Lipschitz property of operators with closed graphs, even in the absence of linearity, thereby extending the classical scope of interpolation theory, which is traditionally confined to linear continuous operators. As a consequence, standard compactness properties are also preserved under mild assumptions. The $R$-method thus provides a new interpolation framework whose foundations rely exclusively on intrinsic metric properties and the summability of discrete orbits, bridging metric interpolation, nonlinear analysis, and classical interpolation theory.