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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2511.22126 |
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| _version_ | 1866911328797458432 |
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| author | Sette, Roblêdo Mak's Miranda |
| author_facet | Sette, Roblêdo Mak's Miranda |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22126 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new interpolation method for metric spaces based on bi-infinite sequences: The $R$-Method Sette, Roblêdo Mak's Miranda Functional Analysis Analysis of PDEs 46B70, 54E35, 47H09 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. |
| title | A new interpolation method for metric spaces based on bi-infinite sequences: The $R$-Method |
| topic | Functional Analysis Analysis of PDEs 46B70, 54E35, 47H09 |
| url | https://arxiv.org/abs/2511.22126 |