The question of $A^*$-summability of double trigonometric Fourier series

It is proved that for any $f(x, y)\in L(R)$, where $R=[-\pi,\pi,-\pi,\pi]$, a function $\varphi(x, y)$, exists such that $|\varphi(x,y)|=|f(x,y)|$ for almost all $(x,y)\in R$. The Fourier series of the function $\varphi(x,y)$ and all conjugate trigonometric series are $A^*$-summable almost everywhere.