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

TL;DR

This paper explores the relationship between the $p$-th variation of continuous functions and $ ext{l}^p$-norms of Schauder basis coefficients, establishing an isomorphism with a class of infinite-dimensional matrices.

Contribution

It introduces a novel isomorphism between spaces of functions with finite $p$-th variation and matrix spaces, extending classical results on H"older continuity.

Findings

Finiteness of $p$-th variation relates to $ ext{l}^p$-norms of basis coefficients.

Established an isomorphism between function spaces and matrix normed spaces.

Connected $p$-th variation properties to Schauder basis coefficient norms.

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 $\ell^p$-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.