Geodesic
Absolute convergence of double series of Fourier--Haar coefficients for functions of bounded $p$-variation
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2012), pp. 3-13

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We consider functions of two variables of bounded $p$-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of the Fourier–Haar series for functions of one variable, provided that they have a bounded Wiener $p$-variation or belong to the class $\operatorname{Lip}\alpha$. We show that the obtained results are unimprovable. We also formulate $N$-dimensional analogs of the main result and its corollaries.
Keywords: double Haar system, functions of two variables of bounded $p$-variation.
Mots-clés : Fourier–Haar coefficients 
@article{IVM_2012_6_a0,
     author = {B. I. Golubov},
     title = {Absolute convergence of double series of {Fourier--Haar} coefficients for functions of bounded $p$-variation},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--13},
     publisher = {mathdoc},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2012_6_a0/}
}
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B. I. Golubov. Absolute convergence of double series of Fourier--Haar coefficients for functions of bounded $p$-variation. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2012), pp. 3-13. http://geodesic.mathdoc.fr/item/IVM_2012_6_a0/