Geodesic
The question of $A^*$-summability of double trigonometric Fourier series
Matematičeskie zametki, Tome 11 (1972) no. 2, pp. 145-150.

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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. 
@article{MZM_1972_11_2_a2,
     author = {L. D. Gogoladze},
     title = {The question of $A^*$-summability of double trigonometric {Fourier} series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {145--150},
     publisher = {mathdoc},
     volume = {11},
     number = {2},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_2_a2/}
}
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L. D. Gogoladze. The question of $A^*$-summability of double trigonometric Fourier series. Matematičeskie zametki, Tome 11 (1972) no. 2, pp. 145-150. http://geodesic.mathdoc.fr/item/MZM_1972_11_2_a2/