Algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces

We prove that the set of all integrable functions whose sequences of negative (resp. nonnegative) Fourier coefficients belong to ${\mathrm{\ell }}^1\cap {\mathrm{\ell }}_{\varphi ,w}^{\mathrm{\Phi }}$ (resp. to ${\mathrm{\ell }}^1\cap {\mathrm{\ell }}_{\psi ,\varrho }^{\mathrm{\Psi }}$), where ${\mathrm{\ell }}_{\varphi ,w}^{\mathrm{\Phi }}$ and ${\mathrm{\ell }}_{\psi ,\varrho }^{\mathrm{\Psi }}$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences $$\varphi =\{{\varphi }_n\},\psi =\{{\psi }_n\},w=\{w_n\},\varrho =\{{\varrho }_n\}$$ increase and satisfy the ${\mathrm{\Delta }}_2$-condition.