On isomorphism of the space of $α$-Hölder continuous functions with finite $p$-th variation

TY - UNPB

T1 - On isomorphism of the space of $α$-Hölder continuous functions with finite $p$-th variation

AU - Das, Purba

AU - Kim, Donghan

PY - 2024/9/1

Y1 - 2024/9/1

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 $\ell^p$-norm of the coefficients along a Schauder basis, similar to the fact that H\"older coefficient of the function is connected to $\ell^{\infty}$-norm of the Schauder coefficients. This result provides an isomorphism between the space of $\alpha$-H\"older 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 $\ell^p$-norm of the coefficients along a Schauder basis, similar to the fact that H\"older coefficient of the function is connected to $\ell^{\infty}$-norm of the Schauder coefficients. This result provides an isomorphism between the space of $\alpha$-H\"older 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.

KW - math.PR

M3 - Preprint

BT - On isomorphism of the space of $α$-Hölder continuous functions with finite $p$-th variation

ER -