On isomorphism of the space of continuous functions with finite $p$-th variation along a partition sequence
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918063447736320 |
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| author | Das, Purba Kim, Donghan |
| author_facet | Das, Purba Kim, Donghan |
| contents | We study the concept of (generalized) $p$-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the $p$-th variation of a given function is closely related to the finiteness of $\ell^p$-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to $\ell^{\infty}$-norm of the Schauder coefficients. This result provides an isomorphism between the space of $α$-Hölder continuous functions with finite (generalized) $p$-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On isomorphism of the space of continuous functions with finite $p$-th variation along a partition sequence Das, Purba Kim, Donghan Probability We study the concept of (generalized) $p$-th variation of a real-valued continuous function along a general class of refining sequence of partitions. We show that the finiteness of the $p$-th variation of a given function is closely related to the finiteness of $\ell^p$-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to $\ell^{\infty}$-norm of the Schauder coefficients. This result provides an isomorphism between the space of $α$-Hölder continuous functions with finite (generalized) $p$-th variation along a given partition sequence and a subclass of infinite-dimensional matrices equipped with an appropriate norm, in the spirit of Ciesielski. |
| title | On isomorphism of the space of continuous functions with finite $p$-th variation along a partition sequence |
| topic | Probability |
| url | https://arxiv.org/abs/2409.00652 |