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 $\ell^1\cap\ell^\Phi_{\phi,w}$ (resp. to $\ell^1\cap\ell^\Psi_{\psi,\varrho}$), where $\ell^\Phi_{\phi,w}$ and $\ell^\Psi_{\psi,\varrho}$ are two-weighted Orlicz sequence spaces, forms an algebra under pointwise multiplication whenever the weight sequences \[ \phi=\{\phi_n\},\quad \psi=\{\psi_n\},\quad w=\{w_n\},\quad \varrho=\{\varrho_n\} \] increase and satisfy the $\Delta_2$-condition.