Summable analogous to the Ivashev-Musatov Theorems

TL;DR

This paper explores summable analogues of the Ivashev-Musatov Theorem within weighted lex^1 and Orlicz sequence spaces, revealing elements with highly pathological support and range, thus extending classical results.

Contribution

It introduces new summable analogues of the Ivashev-Musatov Theorem in advanced sequence spaces, highlighting pathological examples and expanding theoretical understanding.

Findings

Existence of elements with pathological support in weighted lex^1 spaces

Construction of elements with pathological measure-theoretic range

Extension of classical theorems to Orlicz sequence spaces

Abstract

We investigate summable analogues of the classical Ivashev-Musatov Theorem and threshold phenomenons alike. In the setting of weighted $\ell^1$ and Orlicz sequence spaces, we exhibit elements with critically pathological support and range, both topologically and in the sense of measure theory. Our results complement earlier works of J. P. Kahane, Y. Katznelson and T. W. K\"orner.