Functions of bounded variation from ideal perspective
Jacek Gulgowski, Adam Kwela, Jacek Tryba
TL;DR
The paper develops a unifying framework for function spaces of bounded variation via non-pathological lower semicontinuous submeasures $\phi$ on $\mathbb N$, introducing $\mathrm{FIN}(\phi)$, $\mathrm{EXH}(\phi)$, and $\mathrm{BV}(\phi)$ as Banach spaces that generalize classical variation notions. It shows how the Waterman $\Lambda$-variation and Chanturia classes arise as special cases when $\phi$ is chosen to encode summable or simple density behavior, and it develops a two-pronged domination theory (via $\preceq$ and $\preceq_m$) to relate ideal inclusions to BV-inclusions, including a Katětov-order perspective for summable ideals. The work yields a thorough equivalence framework linking submeasures, ideals, and BV-type spaces, and it demonstrates strict separations between the two main special cases, e.g., $\mathrm{BV}(\phi_g)$ typically does not coincide with ${\mathrm{ABV}}$ when growth of $g$ is sufficiently slow. These results provide a conceptual bridge between sequence-space methods and function-variation theory with potential implications for Fourier analysis and approximation contexts.
Abstract
We present a unified approach to two classes of Banach spaces defined with the aid of variations: Waterman spaces and Chanturia classes. Our method is based on some ideas coming from the theory of ideals on the set of natural numbers.