Representation theorems on \(w_ 0(\Phi)\) and \(w_ 0^ 0(\Phi)\) (Q1322697)

Finding representations for continuous linear functionals on function spaces as well as sequence spaces has been the focus of several works in Analysis. In the field of sequence spaces, studies have been made on functionals which are orthogonally additive but not necessarily linear. In particular, the representation theorems for orthogonally additive functionals on the sequence spaces \(\ell_ p\) and \(c_ 0\) were given by \textit{T. S. Chew} and \textit{P. Y. Lee} [ibid. 9, No. 2, 81-85 (1985; Zbl 0613.46007)] and on the space \(w_ 0\) of sequences which are Cesàro strongly summable to zero by \textit{T. S. Chew} [Ann. Soc. Math. Polon., Ser. I, Commentat. Math. 29, No. 2, 149-153 (1990; Zbl 0745.47016)]. In this note, we give a version of the representation theorem for continuous orthogonally additive functionals on the sectionally modulated sequence space \(w_ 0(\Phi)\) and other related spaces.