Geodesic
Absolute convergence of Fourier--Haar series of functions of two variables
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2008), pp. 14-25

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It is well-known that if a one-dimensional function is continuously differentiable on $[0,1]$, then its Fourier–Haar series converges absolutely. On the other hand, if a function of two variables has continuous partial derivatives $f_x'$ and $f_y'$ on $T^2$, then its Fourier series does not necessarily absolutely converge with respect to a multiple Haar system (see [1]). In this paper we state sufficient conditions for the absolute convergence of the Fourier–Haar series for two-dimensional continuously differentiable functions
Keywords: absolute convergence, Fourier series, Haar system, functions of two variables, Rademacher system, convergence almost everywhere. 
@article{IVM_2008_5_a2,
     author = {L. D. Gogoladze and V. Sh. Tsagareishvili},
     title = {Absolute convergence of {Fourier--Haar} series of functions of two variables},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {14--25},
     publisher = {mathdoc},
     number = {5},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_5_a2/}
}
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L. D. Gogoladze; V. Sh. Tsagareishvili. Absolute convergence of Fourier--Haar series of functions of two variables. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2008), pp. 14-25. http://geodesic.mathdoc.fr/item/IVM_2008_5_a2/