Geodesic
Generalized localization for the~multiple Walsh--Fourier series of functions in $L_p$, $p\geqslant 1$
Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 181-196

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In this paper the concept of generalized localization almost everywhere (GL) is analyzed for the multiple Fourier–Walsh series of functions in $L_p(T^N)$, $T^N=[0,1)^N$, $p\geqslant 1$ summable over rectangles. (For multiple trigonometric series and Fourier integrals GL was introduced and analyzed earlier by one of the authors.) If $p>1$, then it is proved that GL holds for double Walsh–Fourier series on any open set. It breaks down on any set $E\subset T^N$ which is not dense in $T^N$ if $N=2$ and $p=1$ and also in the class $\mathbb C$ if $N\geqslant 3$. All results on the Walsh system obtained in this paper are identical to the results on GL for Fourier series in the trigonometric system obtained earlier by one of the authors. 
@article{SM_1995_186_2_a1,
     author = {S. K. Bloshanskaya and I. L. Bloshanskii},
     title = {Generalized localization for the~multiple {Walsh--Fourier} series of functions in $L_p$, $p\geqslant 1$},
     journal = {Sbornik. Mathematics},
     pages = {181--196},
     publisher = {mathdoc},
     volume = {186},
     number = {2},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_2_a1/}
}
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S. K. Bloshanskaya; I. L. Bloshanskii. Generalized localization for the~multiple Walsh--Fourier series of functions in $L_p$, $p\geqslant 1$. Sbornik. Mathematics, Tome 186 (1995) no. 2, pp. 181-196. http://geodesic.mathdoc.fr/item/SM_1995_186_2_a1/