On two variable Beurling algebra analogues of theorems of Wiener and Levy on Fourier series.

Let \omega :\mathbb Z^2 \to [1,\infty) be a weight satisfying a certain technical property. Let f be a continuous function on the torus \mathbb T^2 such that \sum _{\mathbf {m}\in \mathbb Z^2}|\widehat f(\mathbf {m})|\omega (\mathbf {m})<\infty. If f is nowhere vanishing on \mathbb T^2, then there is a weight \nu on \mathbb Z^2 such that 1\leq \nu \leq \omega, \nu is nonconstant if and only if \omega is nonconstant and \sum _{\mathbf {m}\in \mathbb Z^2}\left |\widehat {(\frac {1}{f})}(\mathbf {m})\right |\nu (\mathbf {m})<\infty. If \varphi is holomorphic on a neighbourhood of the range of f, then there is weight \chi on \mathbb Z^2 such that 1\leq \chi \leq \omega, \chi is nonconstant if and only if \omega is nonconstant and \sum _{\mathbf {m}\in \mathbb Z^2}|\widehat {(\varphi \circ f)}(\mathbf {m})|\chi (\mathbf {m})<\infty. An analogue of Żelazko's theorem on p-th power convergence is also discussed. These two variables weighted analogues of classical theorems of Wiener, Lévy and Żelazko are proved exhibiting the complexity of weights in two variables. [ABSTRACT FROM AUTHOR]