Geodesic
On Paley-type theorems for multidimensional Fourier series on generalized Haar-type systems
Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 2, pp. 533-563

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In the paper we prove Paley-type theorems for multidimensional Fourier series on generalized Haar-type systems under the boundedness condition of their generating number sequences. Properties of base of product of generalized Haar-type systems in $L_p[0,1]^n$ are proved. Then, using them, the Paley-type theorems on Fourier coefficients are obtained. Comparisons of our results with the known statements are also given. 
@article{FPM_2001_7_2_a12,
     author = {E. S. Smailov},
     title = {On {Paley-type} theorems for multidimensional {Fourier} series on generalized {Haar-type} systems},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {533--563},
     publisher = {mathdoc},
     volume = {7},
     number = {2},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2001_7_2_a12/}
}
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E. S. Smailov. On Paley-type theorems for multidimensional Fourier series on generalized Haar-type systems. Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 2, pp. 533-563. http://geodesic.mathdoc.fr/item/FPM_2001_7_2_a12/