Geodesic
On an analogue of Hardy's inequality for the Walsh--Fourier
Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 425-435

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According to Hardy's well-known inequality, the $l_1$-norm of a function in the Hardy space $H(T)$ consisting of $2\pi$-periodic functions serves as an upper estimate for the $l_1$-norm of the sequence of Fourier coefficients of the integral of the function. In this paper, the dyadic Hardy space $H(\mathbb R_+)$ is introduced and an analogue of this estimate is proved for the Walsh–Fourier transform. 
@article{IM2_2001_65_3_a0,
     author = {B. I. Golubov},
     title = {On an analogue of {Hardy's} inequality for the {Walsh--Fourier}},
     journal = {Izvestiya. Mathematics },
     pages = {425--435},
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     volume = {65},
     number = {3},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2001_65_3_a0/}
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B. I. Golubov. On an analogue of Hardy's inequality for the Walsh--Fourier. Izvestiya. Mathematics , Tome 65 (2001) no. 3, pp. 425-435. http://geodesic.mathdoc.fr/item/IM2_2001_65_3_a0/