On isomorphism of the space of continuous functions with finite p-th variation along a partition sequence

Purba Das; Donghan Kim

doi:10.1016/j.matpur.2025.103753

On isomorphism of the space of continuous functions with finite p-th variation along a partition sequence

Purba Das
, Donghan Kim

Korea Institute for Advancement of Technology (KIAT)

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Abstract

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 ℓ ^p-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to ℓ ^∞-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.

Original language	English
Article number	103753
Journal	JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
Volume	203
Early online date	16 Jun 2025
DOIs	https://doi.org/10.1016/j.matpur.2025.103753
Publication status	Published - 30 Nov 2025

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math.PR

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AU - Kim, Donghan

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N2 - 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 ℓ p-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to ℓ ∞-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.

AB - 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 ℓ p-norm of the coefficients along a Schauder basis, similar to the fact that Hölder coefficient of the function is connected to ℓ ∞-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.

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ER -

On isomorphism of the space of continuous functions with finite p-th variation along a partition sequence

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