Algebras of functions with Fourier coefficients in weighted Orlicz sequence spaces

Abstract

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.