Gohberg-Krupnik Localisation for Discrete Wiener-Hopf Operators on Orlicz Sequence Spaces

Let $\mathrm{\Phi }$ be an $N$-function whose Matuszewska-Orlicz indices satisfy $1<{\alpha }_{\mathrm{\Phi }}\le {\beta }_{\mathrm{\Phi }}<\mathrm{\infty }$. Using these indices, we introduce ``interpolation friendly" classes of Fourier multipliers $M_{[\mathrm{\Phi }]}$ and $M_{⟨\mathrm{\Phi }⟩}$ such that $M_{[\mathrm{\Phi }]}\subset M_{⟨\mathrm{\Phi }⟩}\subset M_{\mathrm{\Phi }}$, where $M_{\mathrm{\Phi }}$ is the Banach algebra of all Fourier multipliers on the reflexive Orlicz sequence space ${\mathrm{\ell }}^{\mathrm{\Phi }}(\mathbb{Z})$. Applying the Gohberg-Krupnik localisation in the corresponding Calkin algebra, the study of Fredholmness of the discrete Wiener-Hopf operator $T(a)$ with $a\in M_{⟨\mathrm{\Phi }⟩}$ is reduced to that of $T(a_{\tau })$ for certain, potentially easier to study, local representatives $a_{\tau }\in M_{[\mathrm{\Phi }]}$ of $a$ at all points $\tau \in [-\pi ,\pi )$.